Factorisation

-7x² - 14y

= (-7 $\times$ 1) x² + (-7 $\times$ 2) y

= -7(x² + 2y)

Hence, option 4 is the correct option.

a(x - y) - b(x - y)²

= a(x - y) - b(x - y)(x - y)

= (x - y)(a - b(x - y))

= (x - y)(a - bx + by)

Hence, option 3 is the correct option.

a² + bc + ab + ac

= (a x a + ab) + (ac + bc)

= a (a + b) + c(a + b)

= (a + b)(a + c)

Hence, option 1 is the correct option.

1 - 2x - 2x² + 4x³

= 1 $\times$(1 - 2x) - (2x² - 2 $\times$ 2 x² $\times$ x)

= 1 $\times$(1 - 2x) - 2x²(1 - 2x)

= (1 - 2x)(1 - 2x²)

Hence, option 3 is the correct option.

a(x - y) - b(y - x)²

= a(x - y) - b(-(x - y))²

= a(x - y) - b(x - y)²

= (x - y)(a - b(x - y))

= (x - y)(a - bx + by)

Hence, option 4 is the correct option.

17a⁶b⁸ - 34a⁴b⁶ + 51a²b⁴

= 17a⁶b⁸ - 17 x 2a⁴b⁶ + 17 x 3a²b⁴

= 17a²b⁴(a⁴b⁴ - 2a²b² + 3)

Hence, 17a⁶b⁸ - 34a⁴b⁶ + 51a²b⁴ = 17a²b⁴(a⁴b⁴ - 2a²b² + 3)

3x⁵y - 27x⁴y² + 12x³y³

= 3x⁵y - 3 $\times$ 9x⁴y² + 3 $\times$ 4x³y³

= 3x³y(x² - 9xy + 4y²)

Hence, 3x⁵y - 27x⁴y² + 12x³y³ = 3x³y(x² - 9xy + 4y²)

x²(a - b) - y²(a - b) + z²(a - b)

= (a - b)(x² - y² + z²)

Hence,x²(a - b) - y²(a - b) + z²(a - b) = (a - b)(x² - y² + z²)

(x + y)(a + b) + (x - y)(a + b)

= (a + b)$\Big($(x + y) + (x - y)$\Big)$

= (a + b)$\Big($x + y + x - y$\Big)$

= (a + b)(2x)

= 2x(a + b)

Hence, (x + y)(a + b) + (x - y)(a + b) = 2x(a + b)

2b(2a + b) - 3c(2a + b)

= (2a + b)(2b - 3c)

Hence,2b(2a + b) - 3c(2a + b) = (2a + b)(2b - 3c)

12abc - 6a²b²c² + 3a³b³c³

= 3 x 4abc - 3 x 2a²b²c² + 3a³b³c³

= 3abc(4 - 2abc + a²b²c²)

Hence,12abc - 6a²b²c² + 3a³b³c³ = 3abc(4 - 2abc + a²b²c²)

4x(3x - 2y) - 2y(3x - 2y)

= (3x - 2y)(4x - 2y)

= (3x - 2y)2(2x - y)

= 2(3x - 2y)(2x - y)

Hence,4x(3x - 2y) - 2y(3x - 2y) = 2(3x - 2y)(2x - y)

(a + 2b) (3a + b) - (a + b)(a + 2b) + (a + 2b)²

= (a + 2b)$\Big($(3a + b) - (a + b) + (a + 2b)$\Big)$

= (a + 2b)$\Big($ 3a + b - a - b + a + 2b$\Big)$

= (a + 2b) (3a - a + a + b + 2b - b)

= (a + 2b) (3a + 2b)

Hence, (a + 2b) (3a + b) - (a + b)(a + 2b) + (a + 2b)² = (a + 2b) (3a + 2b)

6xy(a² + b²) + 8yz(a² + b²) - 10xz(a² + b²)

= (a² + b²)(6xy + 8yz - 10xz)

= (a² + b²)2(3xy + 4yz - 5xz)

= 2(a² + b²)(3xy + 4yz - 5xz)

Hence, 6xy(a² + b²) + 8yz(a² + b²) - 10xz(a² + b²) = 2(a² + b²)(3xy + 4yz - 5xz)

xy - ay - ax + a² + bx - ab

= y(x - a) - a(x - a) + b(x - a)

= (x - a)(y - a + b)

Hence, xy - ay - ax + a² + bx - ab = (x - a)(y - a + b)

3x⁵ - 6x⁴ - 2x³ + 4x² + x - 2

= 3x⁴(x - 2) - 2x²(x - 2) + 1(x - 2)

= (x - 2)(3x⁴ - 2x² + 1)

Hence,3x⁵ - 6x⁴ - 2x³ + 4x² + x - 2 = (x - 2)(3x⁴ - 2x² + 1)

-x²y - x + 3xy + 3

= - x (xy + 1) + 3(xy + 1)

= (xy + 1)(-x + 3)

= (xy + 1)(3 - x )

Hence, - x²y - x + 3xy + 3 = (xy + 1)(3 - x)

6a² - 3a²b - bc² + 2c²

= 3a²(2 - b) - c²(b - 2)

= 3a²(2 - b) + c²(2 - b)

= (2 - b)(3a² + c²)

Hence,6a² - 3a²b - bc² + 2c² = (2 - b)(3a² + c²)

3a²b - 12a² - 9b + 36

= 3a²(b - 4) - 9(b - 4)

= (b - 4)(3a² - 9)

= (b - 4)3(a² - 3)

= 3(b - 4)(a² - 3)

Hence, 3a²b - 12a² - 9b + 3 = 3(b - 4)(a² - 3)

x² - (a - 3)x - 3a

= x² - ax + 3x - 3a

= x(x - a) + 3(x - a)

= (x - a)(x + 3)

Hence,x² - (a - 3)x - 3a = (x - a)(x + 3)

ab² - (a - c) b - c

= ab² - ab + bc - c

= ab(b - 1) + c(b - 1)

= (b - 1)(ab + c)

Hence,ab² - (a - c) b - c = (b - 1)(ab + c)

(a² - b²) c + (b² - c²)a

= a²c - c²a + b²a - b²c

= ac(a - c) + b²(a - c)

= (a - c)(ac + b²)

Hence,(a² - b²) c + (b² - c²)a = (a - c)(ac + b²)

a³ - a² - ab + a + b - 1

= a³ - a² - ab + b + a - 1

= a²(a - 1) - b(a - 1) + 1(a - 1)

= (a - 1)(a² - b + 1)

Hence,a³ - a² - ab + a + b - 1 = (a - 1)(a² - b + 1)

ab(c² + d²) - a²cd - b²cd

= abc² + abd² - a²cd - b²cd

= (abc² - a²cd) + (abd² - b²cd)

= ac(bc - ad) + bd(ad - bc)

= ac(bc - ad) - bd(bc - ad)

= (bc - ad)(ac - bd)

Hence,ab(c² + d²) - a²cd - b²cd = (bc - ad)(ac - bd)

2ab² - aby + 2cby - cy²

= ab(2b - y) + cy(2b - y)

= (2b - y)(ab + cy)

Hence,2ab² - aby + 2cby - cy² = (2b - y)(ab + cy)

ax + 2bx + 3cx - 3a - 6b - 9c

= x(a + 2b + 3c) - 3(a + 2b + 3c)

= (a + 2b + 3c)(x - 3)

Hence,ax + 2bx + 3cx - 3a - 6b - 9c = (a + 2b + 3c)(x - 3)

2ab²c - 2a + 3b³c - 3b - 4b²c² + 4c

= 2a(b²c - 1) + 3b(b²c - 1) - 4c(b²c - 1)

= (b²c - 1)(2a + 3b - 4c)

Hence,2ab²c - 2a + 3b³c - 3b - 4b²c² + 4c = (b²c - 1)(2a + 3b - 4c)

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

(2x + y)² - (2y + x)²

= ((2x + y) + (2y + x))((2x + y) - (2y + x))

= (2x + y + 2y + x)(2x + y - 2y - x)

= (2x + x + y + 2y)(2x - x + y - 2y)

= (3x + 3y)(x - y)

= 3(x + y)(x - y)

Hence, option 1 is the correct option.

49 - (x + 5)²

= 7² - (x + 5)²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= ((7) + (x + 5))((7) - (x + 5))

= (7 + x + 5)(7 - x - 5)

= (12 + x )(2 - x )

Hence, option 2 is the correct option.

a² - 2ab + b² + a - b

Using the formula

[∵ (x - y)² = x²- 2xy + y² ]

= (a² - 2ab + b²) + a - b

= (a - b)² + 1(a - b)

= (a - b)((a - b) + 1)

= (a - b)(a - b + 1)

Hence, option 4 is the correct option.

x² - y² - 2xy - 1

Using the formula

[∵ (x - y)² = x²- 2xy + y² ]

= (x² + y² - 2xy) - 1

= (x - y)² - 1²

Using the formula

[∵ (x² + y²) = (x + y)(x - y)]

= ((x - y) + (1))((x - y) - (1))

= (x - y + 1)(x - y - 1)

Hence, option 4 is the correct option.

a² + 2a + 1 - b² - x² + 2bx

Using the formula

[∵ (x - y)² = x²- 2xy + y² ]

And

[∵ (x + y)² = x²+ 2xy + y² ]

= (a² + 2a + 1) - (b² + x² - 2bx)

= (a + 1)² - (b - x)²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= ((a + 1) + (b - x))((a + 1) - (b - x))

= (a + 1 + b - x)(a + 1 - b + x)

Hence, option 2 is the correct option.

(a + 2b)² - a²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= ((a + 2b) + a)((a + 2b) - a)

= (a + 2b + a)(a + 2b - a)

= (a + a + 2b)(a - a + 2b)

= (2a + 2b)(2b)

= 2(a + b)2b

= 4b(a + b)

Hence,(a + 2b)² - a² = 4b(a + b)

(5a - 3b)² - 16b²

= (5a - 3b)² - (4b)²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= ((5a - 3b) + 4b)((5a - 3b) - 4b)

= (5a - 3b + 4b)(5a - 3b - 4b)

= (5a + 1b)(5a - 7b)

Hence,(5a - 3b)² - 16b² = (5a + b)(5a - 7b)

a⁴ - (a² - 3b²)²

= (a²)² - (a² - 3b²)²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= ((a²) + (a² - 3b²))((a²) - (a² - 3b²))

= (a² + a² - 3b²)(a² - a² + 3b²)

= (2a²- 3b²)(3b²)

Hence,a⁴ - (a² - 3b²)² = 3b²(2a²- 3b²)

(5a - 2b)² - (2a - b)²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= ((5a - 2b) + (2a - b))((5a - 2b) - (2a - b))

= (5a - 2b + 2a - b)(5a - 2b - 2a + b)

= (5a + 2a - 2b - b)(5a - 2a - 2b + b)

= (7a - 3b)(3a - b)

Hence,(5a - 2b)² - (2a - b)² = (7a - 3b)(3a - b)

1 - 25 (a + b)²

= 1² - (5(a + b))²

= 1² - (5a + 5b)²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= ((1) + (5a + 5b))((1) - (5a + 5b))

= (1 + 5a + 5b)(1 - 5a - 5b)

Hence,1 - 25 (a + b)² = (1 + 5a + 5b)(1 - 5a - 5b)

4(2a + b)² - (a - b)²

= (2(2a + b))² - (a - b)²

= (4a + 2b)² - (a - b)²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= ((4a + 2b) + (a - b))((4a + 2b) - (a - b))

= (4a + 2b + a - b)(4a + 2b - a + b)

= (4a + a + 2b - b)(4a - a + 2b + b)

= (5a+ b)(3a+ 3b)

= (5a+ b)3(a+ b)

Hence,4(2a + b)² - (a - b)² = 3(5a+ b)(a+ b)

25(2x + y)² - 16(x - y)²

= (5(2x + y))² - (4(x - y))²

= (10x + 5y)² - (4x - 4y)²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= ((10x + 5y) + (4x - 4y))((10x + 5y) - (4x - 4y))

= (10x + 5y + 4x - 4y)(10x + 5y - 4x + 4y)

= (10x + 4x + 5y - 4y)(10x - 4x + 5y + 4y)

= (14x + y)(6x + 9y)

= (14x + y)3(2x + 3y)

Hence,25(2x + y)² - 16(x - y)² = 3(14x + y)(2x + 3y)

$\Big(6\frac{2}{3}\Big)^2 - \Big(2\frac{1}{3}\Big)^2$

= $\Big(\frac{20}{3}\Big)^2 - \Big(\frac{7}{3}\Big)^2$

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

$$\begin{aligned}= \Big(\frac{20}{3} + \frac{7}{3}\Big)\Big(\frac{20}{3} - \frac{7}{3}\Big)\\[1em] = \Big(\frac{(20 + 7)}{3}\Big)\Big(\frac{(20 - 7)}{3}\Big)\\[1em] = \Big(\frac{27}{3}\Big)\Big(\frac{13}{3}\Big)\\[1em] = \Big(\frac{27 \times 13}{3 \times 3}\Big)\\[1em] = \Big(\frac{351}{9}\Big)\\[1em] = 39\end{aligned}$$

Hence,$\Big(6\frac{2}{3}\Big)^2 - \Big(2\frac{1}{3}\Big)^2$ = 39

(0.7)² - (0.3)²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= (0.7 + 0.3)(0.7 - 0.3)

= (1.0) x (0.4)

= 0.4

Hence,(0.7)² - (0.3)² = 0.4

75(x + y)² - 48(x - y)²

= 3 x (25(x + y)² - 16(x - y)²)

= 3 x [(5(x + y))² - (4(x - y))²]

= 3 x [(5x + 5y)² - (4x - 4y)²]

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= 3((5x + 5y) + (4x - 4y))((5x + y) - (4x - 4y))

= 3(5x + 5y + 4x - 4y)(5x + 5y - 4x + 4y)

= 3(5x + 4x + 5y - 4y)(5x - 4x + 5y + 4y)

= 3(9x + y)(x + 9y)

Hence,75(x + y)² - 48(x - y)² = 3(9x + y)(x + 9y)

a² + 4a + 4 - b²

Using the formula

[∵ (x + y)² = x² + y² + 2xy]

= (a² + 4a + 4) - b²

= (a + 2)² - b²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= ((a + 2) + b)((a + 2) - b)

= (a + 2 + b)(a + 2 - b)

Hence,a² + 4a + 4 - b² = (a + 2 + b)(a + 2 - b)

a² - b² - 2b - 1

Using the formula

[∵ (x + y)² = x² + y² + 2xy]

= a² - (b² + 2b + 1)

= a² - (b + 1)²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= (a + (b + 1))(a - (b + 1))

= (a + b + 1)(a - b - 1)

Hence,a² - b² - 2b - 1 = (a + b + 1)(a - b - 1)

x² + 6x + 9 - 4y²

Using the formula

[∵ (x + y)² = x² + y² + 2xy]

= (x² + 6x + 9) - 4y²

= (x + 3)² - (2y)²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= ((x + 3) + (2y))((x + 3) - (2y))

= (x + 3 + 2y)(x + 3 - 2y)

Hence,x² + 6x + 9 - 4y² = (x + 3 + 2y)(x + 3 - 2y)

x² - 9x - 10

= x² - (10 - 1)x - 10

= x² - 10x + 1x - 10

= (x² - 10x) + (1x - 10)

= x (x - 10) + 1(x - 10)

= (x - 10)(x + 1)

Hence, option 1 is the correct option.

x² - 23x + 42

= x² - (21 + 2)x + 42

= x² - 21x - 2x + 42

= (x² - 21x) - 2(2x - 42)

= x (x - 21) - 2(x - 21)

= (x - 21)(x - 2)

Hence, option 2 is the correct option.

(4x² - 4x + 1) ÷ (2x - 1)

= $\frac{(4x^2 - 4x + 1)}{(2x - 1)}$

Using the formula,

[∵ (x - y)² = x² + y² - 2xy]

= $\frac{((2x)^2 - 2 \times 2x \times 1 + 1)}{(2x - 1)}$

= $\frac{(2x - 1)^2}{(2x - 1)}$

= $\frac{(2x - 1)(2x - 1)}{(2x - 1)}$

= $\frac{(2x - 1)\cancel{(2x - 1)}}{\cancel{(2x - 1)}}$

= 2x - 1

Hence, option 2 is the correct option.

(x + y)² - 3(x + y) - 4

= (x + y)² + (- 4 + 1)(x + y) - 4

= (x + y)² + (- 4 + 1)(x + y) - 4

= (x + y)² - 4(x + y) + 1(x + y) - 4

= [(x + y)² - 4(x + y)] + [1(x + y) + 4]

= (x + y)[(x + y) - 4] + 1[(x + y) + 4]

= [(x + y) + 1][(x + y) - 4]

= [x + y + 1][x + y - 4]

Hence, option 3 is the correct option.

60 + 11x - x²

= 60 + (15 - 4)x - x²

= 60 + 15x - 4x - x²

= 15(4 + x) - x(4 + x)

= (4 + x)(15 - x)

Hence, option 1 is the correct option.

a² + 5a + 6

= a² + (2 + 3)a + 6

= a² + 2a + 3a + 6

= (a² + 2a) + (3a + 6)

= a(a + 2) + 3(a + 2)

= (a + 2)(a + 3)

Hence,a² + 5a + 6 = (a + 2)(a + 3)

a² - 5a + 6

= a² + (- 2 - 3)a + 6

= a² + - 2a - 3a + 6

= (a² - 2a) - (3a + 6)

= a(a - 2) - 3(a - 2)

= (a - 2)(a - 3)

Hence, a² - 5a + 6 = (a - 2)(a - 3)

a² + 5a - 6

= a² + (6 - 1)a - 6

= a² + 6a - 1a - 6

= (a² + 6a) - (1a + 6)

= a(a + 6) - 1(a + 6)

= (a + 6)(a - 1)

Hence, a² + 5a - 6 = (a + 6)(a - 1)

x² + 5xy + 4y²

= x² + (4 + 1)xy + 4y²

= x² + 4xy + 1xy + 4y²

= (x² + 4xy) + (1xy + 4y²)

= x(x + 4y) + y(x + 4y)

= (x + 4y)(x + y)

Hence,x² + 5xy + 4y² = (x + 4y)(x + y)

a² - 3a - 40

= a² + (- 8 + 5)a - 40

= a² - 8a + 5a - 40

= (a² - 8a) + (5a - 40)

= a(a - 8) + 5(a - 8)

= (a - 8)(a + 5)

Hence,a² - 3a - 40 = (a - 8)(a + 5)

x² - x - 72

= x² + (- 9 + 8)x - 72

= x² - 9x + 8x - 72

= x(x - 9) + 8(x - 9)

= (x - 9)(x + 8)

Hence,x² - x - 72 = (x - 9)(x + 8)

3a² - 5a + 2

= 3a² - (3 + 2)a + 2

= 3a² - 3a - 2a + 2

= (3a² - 3a) - (2a - 2)

= 3a(a - 1) - 2(a - 1)

= (a - 1)(3a - 2)

Hence,3a² - 5a + 2 = (a - 1)(3a - 2)

2a² - 17ab + 26b²

= 2a² - (13 + 4)ab + 26b²

= 2a² - 13ab - 4ab + 26b²

= (2a² - 13ab) - (4ab - 26b²)

= a(2a - 13b) - 2b(2a - 13b)

= (2a - 13b)(a - 2b)

Hence,2a² - 17ab + 26b² = (2a - 13b)(a - 2b)

2x² + xy - 6y²

= 2x² + (4 - 3)xy - 6y²

= 2x² + 4xy - 3xy - 6y²

= (2x² + 4xy) - (3xy + 6y²)

= 2x(x + 2y) - 3y(x + 2y)

= (x + 2y)(2x - 3y)

Hence,2x² + xy - 6y² = (x + 2y)(2x - 3y)

4c² + 3c - 10

= 4c² + (8 - 5)c - 10

= 4c² + 8c - 5c - 10

= (4c² + 8c) - (5c + 10)

= 4c(c + 2) - 5(c + 2)

= (c + 2)(4c - 5)

Hence,4c² + 3c - 10 = (c + 2)(4c - 5)

14x² + x - 3

= 14x² + (7 - 6)x - 3

= 14x² + 7x - 6x - 3

= 7x(2x + 1) - 3(2x + 1)

= (2x + 1)(7x - 3)

Hence,14x² + x - 3 = (2x + 1)(7x - 3)

6 + 7b - 3b²

= 6 + (9 - 2)b - 3b²

= 6 + 9b - 2b - 3b²

= (6 + 9b) - (2b + 3b²)

= 3(2 + 3b) - b(2 + 3b)

= (2 + 3b)(3 - b)

Hence,6 + 7b - 3b² = (2 + 3b)(3 - b)

5 + 7x - 6x²

= 5 + (10 - 3)x - 6x²

= 5 + 10x - 3x - 6x²

= (5 + 10x) - (3x + 6x²)

= 5(1 + 2x) - 3x(1 + 2x)

= (1 + 2x)(5 - 3x)

Hence,5 + 7x - 6x² = (1 + 2x)(5 - 3x)

4 + y - 14y²

= 4 + (8 - 7)y - 14y²

= 4 + 8y - 7y - 14y²

= (4 + 8y) - (7y + 14y²)

= 4(1 + 2y) - 7y(1 + 2y)

= (1 + 2y)(4 - 7y)

Hence,4 + y - 14y² = (1 + 2y)(4 - 7y)

5 + 3a - 14a²

= 5 + (10 - 7)a - 14a²

= 5 + 10a - 7a - 14a²

= (5 + 10a) - (7a + 14a²)

= 5(1 + 2a) - 7a(1 + 2a)

= (1 + 2a)(5 - 7a)

Hence,5 + 3a - 14a² = (1 + 2a)(5 - 7a)

(2a + b)² + 5(2a + b) + 6

= (2a + b)² + (2 + 3)(2a + b) + 6

= (2a + b)² + 2(2a + b) + 3(2a + b) + 6

= [(2a + b)² + 2(2a + b)] + [3(2a + b) + 6]

= (2a + b)[(2a + b) + 2] + 3[(2a + b) + 2]

= [(2a + b) + 2][(2a + b) + 3]

= [2a + b + 2][2a + b + 3]

Hence,(2a + b)² + 5(2a + b) + 6 = (2a + b + 2)(2a + b + 3)

1 - (2x + 3y) - 6(2x + 3y)²

= 1 - (3 - 2)(2x + 3y) - 6(2x + 3y)²

= 1 - 3(2x + 3y) + 2(2x + 3y) - 6(2x + 3y)²

= 1[1 - 3(2x + 3y)] + 2(2x + 3y)[1 - 3(2x + 3y)]

= [1 - 3(2x + 3y)][1 + 2(2x + 3y)]

= [1 - 6x - 9][1 + 4x + 6y]

Hence,1 - (2x + 3y) - 6(2x + 3y)² = [1 - 6x - 9y][1 + 4x + 6y]

(x - 2y)² - 12(x - 2y) + 32

= (x - 2y)² - (8 + 4)(x - 2y) + 32

= (x - 2y)² - 8(x - 2y) - 4(x - 2y) + 32

= [(x - 2y)² - 8(x - 2y)] - [4(x - 2y) - 32]

= (x - 2y)[(x - 2y) - 8] - 4[(x - 2y) - 8]

= [(x - 2y) - 8][(x - 2y) - 4]

= [x - 2y - 8][x - 2y - 4]

Hence,(x - 2y)² - 12(x - 2y) + 32 = [x - 2y - 8][x - 2y - 4]

8 + 6(a + b) - 5(a + b)²

= 8 + (10 - 4)(a + b) - 5(a + b)²

= 8 + 10(a + b) - 4(a + b) - 5(a + b)²

= [8 + 10(a + b)] - [4(a + b) + 5(a + b)²]

= 2[4 + 5(a + b)] - (a + b)[4 + 5(a + b)]

= [4 + 5(a + b)][2 - (a + b)]

= [4 + 5a + 5b)][2 - a - b]

Hence,8 + 6(a + b) - 5(a + b)² = [4 + 5a + 5b][2 - a - b]

2(x + 2y)² - 5(x + 2y) + 2

= 2(x + 2y)² - (4 + 1)(x + 2y) + 2

= 2(x + 2y)² - 4(x + 2y) - 1(x + 2y) + 2

= [2(x + 2y)² - 4(x + 2y)] - [1(x + 2y) - 2]

= 2(x + 2y)[(x + 2y) - 2] - 1[(x + 2y) - 2]

= [(x + 2y) - 2][2(x + 2y) - 1]

= [x + 2y - 2][2x + 4y - 1]

Hence,2(x + 2y)² - 5(x + 2y) + 2 = [x + 2y - 2][2x + 4y - 1]

x² + 14x + 49

Using the formula

[∵ (x + y)² = x²+ 2xy + y² ]

= x² + 2 $\times$ 7 $\times$ x + (7)²

= (x + 7)²

Hence, x² + 14x + 49 is a perfect square.

a² - 10a + 25

Using the formula

[∵ (x - y)² = x²- 2xy + y² ]

= a² - 2 $\times$ a $\times$ 5 + (5)²

= (a + 5)²

Hence, a² - 10a + 25 is a perfect square.

4x² + 4x + 1

Using the formula

[∵ (x + y)² = x²+ 2xy + y² ]

= (2x)² + 2 $\times$ 2x $\times$ 1 + (1)²

= (2x + 1)²

Hence, 4x² + 4x + 1 is a perfect square.

9b² + 12b + 16

Using the formula

[∵ (x + y)² = x²+ 2xy + y² ]

Since, (9b² + 12b + 16) cannot be expressed as a² + 2ab + b².

Hence, 9b² + 12b + 16 is not a perfect square.

16x² - 16xy + y²

Using the formula

[∵ (x - y)² = x²- 2xy + y² ]

Since, (16x² - 16xy + y²) cannot be expressed as a² - 2ab + b².

Hence, 16x² - 16xy + y² is not a perfect square.

x² - 4x + 16

Using the formula

[∵ (x - y)² = x²- 2xy + y² ]

Since, (x² - 4x + 16) cannot be expressed as a² - 2ab + b².

Hence, x² - 4x + 16 is not a perfect square.

x³ - 4x

= x (x² - 4)

= x ((x)² - (2)²)

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= x (x + 2)(x - 2)

Hence, option 2 is the correct option.

x⁴ - y⁴ + x² - y²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= (x⁴ - y⁴) + (x² - y²)

= (x² - y²)(x² + y²) + (x - y)(x + y)

= (x - y)(x + y)(x² + y²) + (x - y)(x + y)

= (x - y)(x + y)((x² + y²) + 1)

= (x - y)(x + y)(x² + y² + 1)

Hence, option 3 is the correct option.

x³ - x² + ax + x - a - 1

= (x³ - x²) + (ax - a) + (x - 1)

= x²(x - 1) + a(x - 1) + 1(x - 1)

= (x - 1)(x² + a + 1)

Hence, option 2 is the correct option.

8x³ - 18x

= 2x (4x² - 9)

= 2x ((2x)² - (3)²)

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= 2x (2x - 3)(2x + 3)

Hence, option 3 is the correct option.

x² - (a - b) x - ab

= x² - ax + bx - ab

= (x² - ax) + (bx - ab)

= x(x - a) + b(x - a)

= (x - a)(x + b)

Hence, option 3 is the correct option.

8x²y - 18y³

= 2y (4x² - 9y²)

= 2y ((2x)² - (3y)²)

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= 2y (2x - 3y)(2x + 3y)

Hence,8x²y - 18y³ = 2y (2x - 3y)(2x + 3y)

25x³ - x

= x (25x² - 1)

= x ((5x)² - (1)²)

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= x (5x + 1)(5x - 1)

Hence,25x³ - x = x (5x + 1)(5x - 1)

16x⁴ - 81y⁴

= (4x²)² - (9y²)²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= (4x² + 9y²)(4x² - 9y²)

= (4x² + 9y²)((2x)² - (3y)²)

= (4x² + 9y²)(2x - 3y)(2x + 3y)

Hence,16x⁴ - 81y⁴ = (4x² + 9y²)(2x - 3y)(2x + 3y)

x² - y² - 3x - 3y

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= (x² - y²) - (3x + 3y)

= (x - y)(x + y) - 3(x + y)

= (x + y)((x - y) - 3)

= (x + y)(x - y - 3)

Hence,x² - y² - 3x - 3y = (x + y)(x - y - 3)

x² - y² - 2x + 2y

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= (x² - y²) - (2x - 2y)

= (x - y)(x + y) - 2(x - y)

= (x - y)((x + y) - 2)

= (x - y)(x + y - 2)

Hence,x² - y² - 2x + 2y = (x - y)(x + y - 2)

3x² + 15x - 72

= 3(x² + 5x - 24)

= 3(x² + (8 - 3)x - 24)

= 3(x² + 8x - 3x - 24)

= 3[(x² + 8x) - (3x + 24)]

= 3[x(x + 8) - 3(x + 8)]

= 3(x + 8)(x - 3)

Hence,3x² + 15x - 72 = 3(x + 8)(x - 3)

2a² - 8a - 64

= 2(a² - 4a - 32)

= 2(a² - (8 - 4)a - 32)

= 2(a² - 8a + 4a - 32)

= 2[(a² - 8a) + (4a - 32)]

= 2[a(a - 8) + 4(a - 8)]

= 2(a - 8)(a + 4)

Hence,2a² - 8a - 64 = 2(a - 8)(a + 4)

3x²y + 11xy + 6y

= 3x²y + (9 + 2)xy + 6y

= 3x²y + 9xy + 2xy + 6y

= (3x²y + 9xy) + (2xy + 6y)

= 3xy(x + 3) + 2y(x + 3)

= (x + 3)(3xy + 2y)

= (x + 3)y(3x + 2)

Hence,3x²y + 11xy + 6y = y(x + 3)(3x + 2)

5ap² + 11ap + 2a

= a(5p² + 11p + 2)

= a(5p² + (10 + 1)p + 2)

= a(5p² + 10p + 1p + 2)

= a[(5p² + 10p) + (1p + 2)]

= a[5p(p + 2) + 1(p + 2)]

= a(p + 2)(5p + 1)

Hence,5ap² + 11ap + 2a = a(p + 2)(5p + 1)

a² + 2ab + b² - c²

Using the formula

[∵ (x + y)² = x² + y² + 2xy]

= (a² + 2ab + b²) - c²

= ((a)² + 2 $\times$ a $\times$ b + (b)²) - c²

= (a + b)² - (c)²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= ((a + b) + c)((a + b) - c)

= (a + b + c)(a + b - c)

Hence,a² + 2ab + b² - c² = (a + b + c)(a + b - c)

x² + 6xy + 9y² + x + 3y

Using the formula

[∵ (x + y)² = x² + y² + 2xy]

= (x² + 6xy + 9y²) + x + 3y

= ((x)² + 2 $\times$ x $\times$ 3y + (3y)²) + x + 3y

= (x + 3y)² + (x + 3y)

= (x + 3y)((x + 3y) + 1)

= (x + 3y)(x + 3y + 1)

Hence,x² + 6xy + 9y² + x + 3y = (x + 3y)(x + 3y + 1)

4a² - 12ab + 9b² + 4a - 6b

Using the formula

[∵ (x - y)² = x² + y² - 2xy]

= (4a² - 12ab + 9b²) + 4a - 6b

= ((2a)² - 2 $\times$ 2a $\times$ 3b + (3b)²) + 4a - 6b

= (2a - 3b)² + (4a - 6b)

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= (2a - 3b)² + 2(2a - 3b)

= (2a - 3b)((2a - 3b) + 2)

= (2a - 3b)(2a - 3b + 2)

Hence,4a² - 12ab + 9b² + 4a - 6b = = (2a - 3b)(2a - 3b + 2)

2a²b² - 98b⁴

= 2b²(a² - 49b²)

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= 2b²((a)² - (7b)²)

= 2b²(a - 7b)(a + 7b)

Hence,2a²b² - 98b⁴ = 2b²(a - 7b)(a + 7b)

a² - 16b² - 2a - 8b

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= (a² - 16b²) - (2a + 8b)

= (a - 4b)(a + 4b) - 2(a + 4b)

= (a + 4b)((a - 4b) - 2)

= (a + 4b)(a - 4b - 2)

Hence,a² - 16b² - 2a - 8b = (a + 4b)(a - 4b - 2)

(a + b)² - 4ab

Using the formula

[∵ (x + y)² = x² + y² + 2xy]

= a² + b² + 2ab - 4ab

= a² + b² + (2ab - 4ab)

= a² + b² - 2ab

Using the formula

[∵ (x + y)² = x² + y² + 2xy]

= (a)² + (b)² - 2 x a x b

= (a - b)²

= (a - b)(a - b)

Hence, option 4 is the correct option.

a⁴ + 4a² - 32

= a⁴ + (8 - 4)a² - 32

= a⁴ + 8a² - 4a² - 32

= (a⁴ + 8a²) - (4a² + 32)

= a²(a² + 8) - 4(a² + 8)

= (a² + 8)(a² - 4)

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= (a² + 8)(a - 2)(a + 2)

Hence, option 4 is the correct option.

36 - 60y + 25y²

Using the formula

[∵ (x + y)² = x² + y² + 2xy]

= (6)² - 2 x 6 x 5y + (5y)²

= (6 - 5y)²

= (6 - 5y)(6 - 5y)

Hence, option 4 is the correct option.

(x - 2y)² - 3x + 6y

= (x - 2y)² - 3(x - 2y)

= (x - 2y)((x - 2y) - 3)

= (x - 2y)(x - 2y - 3)

Hence, option 4 is the correct option.

a(x - y)² - by + bx

= a(x - y)² - (by - bx)

= a(x - y)² - b(y - x)

= a(x - y)² + b(x - y)

= (x - y)(a(x - y) + b)

= (x - y)(ax - ay + b)

Hence, option 4 is the correct option.

According to statement 1 :

The product of two binomials is a trinomial.

For example; (x + 2)(x + 3) = x(x + 3) + 2(x + 3)

= x² + 3x + 2x + 6

= x² + 5x + 6

Here, (x + 2) and (x + 3) are two binomials and their product x² + 5x + 6 is a trinomial.

But, (x + 2)(x - 2) = x² - 2²

= x² - 4

Here, (x + 2) and (x - 2) are two binomials but x² - 4 is not a trinomial.

So, we can say that the product of two binomials is not necessarily a trinomial.

So, statement 1 is true.

The difference of the square of two terms = the sum of the same two terms x their difference.

As,

a² - b² =(a - b) (a + b)

L.H.S. = a² - b²

R.H.S. = (a - b) x (a + b)

Therefore, L.H.S. = R.H.S.

$\Rightarrow$ The difference of the square of two terms = the sum of the same two terms x their difference.

So, statement 2 is also true.

Hence, option 1 is the correct option.

(5x - 1)² = (5x)² - 2.5x.1 + 1²

= 25x² - 10x + 1²

Thus, 25x² - 5x + 1 is not a perfect square trinomial.

So, assertion (A) is false.

(x - y)² = x² - 2.x.y + y²

= x² - 2xy + y²

And, (x + y)² = x² + 2.x.y + y²

= x² + 2xy + y²

Thus, x² + y² + 2xy or x² + y² - 2xy is a perfect square trinomial.

So, reason (R) is true.

Hence, option 4 is the correct option.

For a quadratic trinomial of the form :

ax² + bx + c

To factor using the middle-term splitting method;

Find two numbers that multiply to give ac and add to give b.

So, reason (R) is true.

Given; x² + 7x + 12

= x² + (3 + 4)x + 4 x 3

= x² + 3x + 4x + 4 x 3

= x(x + 3) + 4(x + 3)

= (x + 3)(x + 4)

So, assertion (A) is true and, reason (R) is the correct explanation of assertion (A).

Hence, option 1 is the correct option.

Given; $\Big(x^2 - kx + \frac{121}{16}\Big)$

We know that this quadratic has equal factors if it's a perfect square trinomial, meaning :

$$\begin{aligned}\Rightarrow \Big(x^2 - kx + \frac{121}{16}\Big) = \Big(x - \frac{11}{4}\Big)^2 \\[1em] \Rightarrow \Big(x^2 - kx + \frac{121}{16}\Big) = x^2 - 2 \times x \times \frac{11}{4} + \Big(\frac{11}{4}\Big)^2 \\[1em] \Rightarrow \Big(x^2 - kx + \frac{121}{16}\Big) = x^2 - \frac{11}{2}x + \frac{121}{16} \\[1em] \Rightarrow k = \frac{11}{2}.\end{aligned}$$

So, assertion (A) is true.

Solving,

(x + a)(x + b) = x(x + b) + a(x + b)

= x² + bx + ax + ab

= x² + x(a + b) + ab

So, reason (R) is true but, reason (R) is not the correct explanation of assertion (A).

Hence, option 2 is the correct option.

Given :

x² + bx - 24

To factor such a quadratic, we look for two numbers that :

Product = -24 and Sum = b

These values can be 6 and -4, -6 and 4, 1 and -24, -1 and 24, -3 and 8, -8 and 3, 12 and -2, 2 and -12.

In these cases value of b can be : 6 + (-4) = 2, (-6) + 4 = -2, 1 + (-24) = -23, (-1) + 24 = 23, (-3) + 8 = 5, (-8) + 3 = -5, (-12) + 2 = -10, 12 + (-2) = 10.

Thus, there are more than 2 values of b for x² + bx - 24 to be factorisable.

∴ Assertion (A) is false.

Values have :

Product = -24 and sum = 2 or -2, 23 or -23, 10 or -10, 5 or -5.

∴ Reason (R) is false.

Hence, Both Assertion (A) and Reason (R) are false.

6x³ - 8x²

= 2x² (3x - 4)

Hence,6x³ - 8x² = 2x² (3x - 4)

36x²y² - 30x³y³ + 48x³y²

= 6x²y² (6 - 5xy + 8x)

Hence,36x²y² - 30x³y³ + 48x³y² = 6x²y² (6 - 5xy + 8x)

8(2a + 3b)³ - 12(2a + 3b)²

= 4(2(2a + 3b)³ - 3(2a + 3b)²)

= 4(2a + 3b)²(2(2a + 3b) - 3)

= 4(2a + 3b)²(4a + 6b - 3)

Hence,8(2a + 3b)³ - 12(2a + 3b)² = 4(2a + 3b)²(4a + 6b - 3)

9a(x - 2y)⁴ - 12a(x - 2y)³

= 3a(3(x - 2y)⁴ - 4(x - 2y)³)

= 3a(x - 2y)³(3(x - 2y) - 4)

= 3a(x - 2y)³(3x - 6y - 4)

Hence,9a(x - 2y)⁴ - 12a(x - 2y)³ = 3a(x - 2y)³(3x - 6y - 4)

a² - ab(1 - b) - b³

= a² - ab + ab² - b³

= (a² - ab) + (ab² - b³)

= a(a - b) + b²(a - b)

= (a - b)(a + b²)

Hence,a² - ab(1 - b) - b³ = (a - b)(a + b²)

xy² + (x - 1)y - 1

= xy² + xy - y - 1

= (xy² + xy) - (y + 1)

= xy(y + 1) - 1(y + 1)

= (y + 1)(xy - 1)

Hence,xy² + (x - 1)y - 1 = (y + 1)(xy - 1)

(ax + by)² + (bx - ay)²

Using the formula

[∵ (x + y)² = x² + y² + 2xy]

= a²x² + b²y² + 2abxy + b²x² + a²y² - 2abxy

= (a²x² + a²y²) + (b²y² + b²x²) + 2abxy- 2abxy

= a²(x² + y²) + b²(x² + y²)

= (x² + y²)(a² + b²)

Hence,(ax + by)² + (bx - ay)² = (x² + y²)(a² + b²)

ab(x² + y²) - xy(a² + b²)

= abx² + aby² - xya² - xyb²

= (abx² - xya²) + (aby² - xyb²)

= (abx² - a²xy) + (aby² - b²xy)

= ax(bx - ay) + by(ay - bx)

= ax(bx - ay) - by(bx - ay)

= (bx - ay)(ax - by)

Hence,ab(x² + y²) - xy(a² + b²) = (bx - ay)(ax - by)

m - 1 - (m - 1)² + am - a

= (m - 1) - (m - 1)² + (am - a)

= 1(m - 1) - (m - 1)² + a(m - 1)

= (m - 1)(1 - (m - 1) + a)

= (m - 1)(1 - m + 1 + a)

= (m - 1)(2 - m + a)

Hence,m - 1 - (m - 1)² + am - a = (m - 1)(2 - m + a)

25(2x - y)² - 16(x - 2y)²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= (5(2x - y))² - (4(x - 2y))²

= (5(2x - y) + 4(x - 2y))(5(2x - y) - 4(x - 2y))

= (10x - 5y + 4x - 8y)(10x - 5y - 4x + 8y)

= (10x + 4x - 5y - 8y)(10x - 4x - 5y + 8y)

= (14x - 13y)(6x + 3y)

= (14x - 13y)3(2x + y)

Hence,25(2x - y)² - 16(x - 2y)² = 3(14x - 13y)(2x + y)

16(5x + 4)² - 9(3x - 2)²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= (4(5x + 4))² - (3(3x - 2))²

= (4(5x + 4) + 3(3x - 2))(4(5x + 4) - 3(3x - 2))

= (20x + 16 + 9x - 6)(20x + 16 - 9x + 6)

= (20x + 9x + 16 - 6)(20x - 9x + 16 + 6)

= (29x + 10)(11x + 22y)

= (29x + 10)11(x + 2y)

Hence,16(5x + 4)² - 9(3x - 2)² = 11(29x + 10)(x + 2y)

25(x - 2y)² - 4

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= (5(x - 2y))² - (2)²

= (5(x - 2y) - 2)(5(x - 2y) + 2)

= (5x - 10y - 2)(5x - 10y + 2)

Hence,25(x - 2y)² - 4 = (5x - 10y - 2)(5x - 10y + 2)

a² - 23a + 42

= a² - (21 + 2)a + 42

= a² - 21a - 2a + 42

= (a² - 21a) - (2a - 42)

= a(a - 21) - 2(a - 21)

= (a - 21)(a - 2)

Hence,a² - 23a + 42 = (a - 21)(a - 2)

a² - 23a - 108

= a² - 23a - 108

= a² - (27 - 4)a - 108

= a² - 27a + 4a - 108

= (a² - 27a) + (4a - 108)

= a(a - 27) + 4(a - 27)

= (a - 27)(a + 4)

Hence,a² - 23a - 108 = (a - 27)(a + 4)

1 - 18x - 63x²

= 1 - (21 - 3)x - 63x²

= 1 - 21x + 3x - 63x²

= (1 - 21x) + (3x - 63x²)

= 1(1 - 21x) + 3x(1 - 9x)

= (1 - 21x)(1 + 3x)

Hence,1 - 18x - 63x² = (1 - 21x)(1 + 3x)

5x² - 4xy - 12y²

= 5x² - (10 - 6)xy - 12y²

= 5x² - 10xy + 6xy - 12y²

= (5x² - 10xy) + (6xy - 12y²)

= 5x(x - 2y) + 6y(x - 2y)

= (x - 2y)(5x + 6y)

Hence,5x² - 4xy - 12y² = (x - 2y)(5x + 6y)

x(3x + 14) + 8

= 3x² + 14x + 8

= 3x² + (12 + 2)x + 8

= 3x² + 12x + 2x + 8

= (3x² + 12x) + (2x + 8)

= 3x(x + 4) + 2(x + 4)

= (x + 4)(3x + 2)

Hence,x(3x + 14) + 8 = (x + 4)(3x + 2)

5 - 4x(1 + 3x)

= 5 - 4x - 12x²

= 5 - (10 - 6)x - 12x²

= 5 - 10x + 6x - 12x²

= (5 - 10x) + (6x - 12x²)

= 5(1 - 2x) + 6x(1 - 2x)

= (1 - 2x)(5 + 6x)

Hence,5 - 4x(1 + 3x) = (1 - 2x)(5 + 6x)

x²y² - 3xy - 40

= x²y² - (8 - 5)xy - 40

= x²y² - 8xy + 5xy - 40

= (x²y² - 8xy) + (5xy - 40)

= xy(xy - 8) + 5(xy - 8)

= (xy - 8)(xy + 5)

Hence, x²y² - 3xy - 40 = (xy - 8)(xy + 5)

(3x - 2y)² - 5(3x - 2y) - 24

= (3x - 2y)² - (8 - 3)(3x - 2y) - 24

= (3x - 2y)² - 8(3x - 2y) + 3(3x - 2y) - 24

= (3x - 2y)((3x - 2y) - 8) + 3((3x - 2y) - 8)

= (3x - 2y)(3x - 2y - 8) + 3(3x - 2y - 8)

= (3x - 2y - 8)(3x - 2y + 3)

Hence,(3x - 2y)² - 5(3x - 2y) - 24 = (3x - 2y - 8)(3x - 2y + 3)

12(a + b)² - (a + b) - 35

= 12(a + b)² - (21 - 20)(a + b) - 35

= 12(a + b)² - 21(a + b) + 20(a + b) - 35

= 3(a + b)(4(a + b) - 7) + 5(4(a + b) - 7)

= 3(a + b)(4a + 4b - 7) + 5(4a + 4b - 7)

= (4a + 4b - 7)(3(a + b) + 5)

= (4a + 4b - 7)(3a + 3b + 5)

Hence,12(a + b)² - (a + b) - 35 = (4a + 4b - 7)(3a + 3b + 5)

15(5x - 4)² - 10(5x - 4)

= 5(5x - 4)(3(5x - 4) - 2)

= 5(5x - 4)(15x - 12 - 2)

= 5(5x - 4)(15x - 14)

Hence,15(5x - 4)² - 10(5x - 4) = 5(5x - 4)(15x - 14)

3a²x - bx + 3a² - b

= x(3a² - b) + 1(3a² - b)

= (3a² - b)(x + 1)

Hence,3a²x - bx + 3a² - b = (3a² - b)(x + 1)

b(c - d)² + a(d - c) + 3(c - d)

= b(c - d)² - a(c - d) + 3(c - d)

= (c - d)(b(c - d) - a + 3)

= (c - d)(bc - bd - a + 3)

Hence,b(c - d)² + a(d - c) + 3(c - d) = (c - d)(bc - bd - a + 3)

ax² + b²y - ab² - x²y

= (ax² - ab²) + (b²y - x²y)

= a (x² - b²) + y(b² - x²)

= a (x² - b²) - y(x² - b²)

= (x² - b²)(a - y)

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= (x - b)(x + b)(a - y)

Hence,ax² + b²y - ab² - x²y = (x - b)(x + b)(a - y)

1 - 3x - 3y - 4(x + y)²

= 1 - 3(x + y) - 4(x + y)²

= 1 - (4 - 1)(x + y) - 4(x + y)²

= 1 - 4(x + y) + 1(x + y) - 4(x + y)²

= (1 - 4(x + y)) + (1(x + y) - 4(x + y)²)

= 1(1 - 4(x + y)) + (x + y)(1 - 4(x + y))

= 1(1 - 4x - 4y) + (x + y)(1 - 4x - 4y)

= (1 - 4x - 4y)(1 + (x + y))

= (1 - 4x - 4y)(1 + x + y)

Hence,1 - 3x - 3y - 4(x + y)² = (1 - 4x - 4y)(1 + x + y)

2a³ - 50a

= 2a(a² - 25)

= 2a(a² - 5²)

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= 2a(a - 5)(a + 5)

Hence,2a³ - 50a = 2a(a - 5)(a + 5)

54a²b² - 6

= 6(9a²b² - 1)

= 6((3ab)² - (1)²)

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= 6(3ab - 1)(3ab + 1)

Hence, 54a²b² - 6 = 6(3ab - 1)(3ab + 1)

64a²b - 144b³

= 16b(4a² - 9b²)

= 16b((2a)² - (3b)²)

= 16b(2a + 3b)(2a - 3b)

Hence, 64a²b - 144b³ = 16b(2a + 3b)(2a - 3b)

(2x - y)³ - (2x - y)

= (2x - y)((2x - y)² - 1)

= (2x - y)((2x - y)² - (1)²)

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= (2x - y)((2x - y) - 1)((2x - y) + 1)

= (2x - y)(2x - y - 1)(2x - y + 1)

Hence, (2x - y)³ - (2x - y) = (2x - y)(2x - y - 1)(2x - y + 1)

x² - 2xy + y² - z²

Using the formula

[∵ (x - y)² = x² + y² - 2xy]

= (x² - 2xy + y²) - z²

= (x - y)² - z²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= ((x - y) - z)((x - y) + z)

= (x - y - z)(x - y + z)

Hence, x² - 2xy + y² - z² = (x - y - z)(x - y + z)

x² - y² - 2yz - z²

Using the formula

[∵ (x - y)² = x² + y² - 2xy]

= x² - (y² + 2yz + z²)

= x² - (y + z)²

= (x)² - (y + z)²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= (x - (y + z))(x + (y + z))

= (x - y - z)(x + y + z)

Hence, x² - y² - 2yz - z² = (x - y - z)(x + y + z)

7a⁵ - 567a

= 7a(a⁴ - 81)

= 7a((a²)⁴ - (9)⁴)

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= 7a(a² - 9)(a² + 9)

= 7a((a)² - (3)²)(a² + 9)

= 7a(a - 3)(a + 3)(a² + 9)

Hence, 7a⁵ - 567a = 7a(a - 3)(a + 3)(a² + 9)

$$\begin{aligned}5x^2 - \frac{20x^4}{9}\\[1em] = 5x^2\Big(1 - \frac{4x^2}{9}\Big)\\[1em] = 5x^2\Big(1^2 - \Big(\frac{2x}{3}\Big)^2\Big)\\[1em]\end{aligned}$$

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

$= 5x^2\Big(1 - \frac{2x}{3}\Big)\Big(1 + \frac{2x}{3}\Big)$

Hence, $5x^2 - \frac{20x^4}{9} = 5x^2\Big(1 - \frac{2x}{3}\Big)\Big(1 + \frac{2x}{3}\Big)$

xy² - xz²

= x(y² - z²)

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= x(y - z)(y + z)

Hence, xy² - xz² = x(y - z)(y + z)

(i) 9 x 8² - 9 x 2²

= 9 x (8 - 2)(8 + 2)

= 9 x 6 x 10

= 540

Hence, 9 x 8² - 9 x 2² = 540

(ii) 40 x 5.5² - 40 x 4.5²

= 40 x (5.5 - 4.5) x (5.5 + 4.5)

= 40 x 1 x 10

= 400

Hence, 40 x 5.5² - 40 x 4.5² = 400

(a - 3b)² - 36 b²

= (a - 3b)² - (6 b)²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= ((a - 3b) - 6b)((a - 3b) + 6b)

= (a - 3b - 6b)(a - 3b + 6b)

= (a - 9b)(a + 3b)

Hence, (a - 3b)² - 36 b² = (a - 9b)(a + 3b)

25(a - 5b)² - 4(a - 3b)²

= (5(a - 5b))² - (2(a - 3b))²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= (5(a - 5b) - 2(a - 3b))((5(a - 5b)) + 2(a - 3b))

= (5a - 25b - 2a + 6b)(5a - 25b + 2a - 6b)

= (5a - 2a - 25b + 6b)(5a + 2a - 25b - 6b)

= (3a - 19b)(7a - 31b)

Hence, 25(a - 5b)² - 4(a - 3b)² = (3a - 19b)(7a - 31b)

a² - 0.36 b²

= (a)² - (0.6 b)²

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= (a + 0.6b)(a - 0.6b)

Hence, a² - 0.36 b² = (a + 0.6b)(a - 0.6b)

x⁴ - 5x² - 36

= x⁴ - (9 - 4)x² - 36

= x⁴ - 9x² + 4x² - 36

= (x⁴ - 9x²) + (4x² - 36)

= x²(x² - 9) + 4(x² - 9)

= (x² - 9)(x² + 4)

= ((x)² - (3)²)(x² + 4)

= (x - 3)(x + 3)(x² + 4)

Hence, x⁴ - 5x² - 36 = (x - 3)(x + 3)(x² + 4)

15(2x - y)² - 16(2x - y) - 15

= 15(2x - y)² - (25 - 9)(2x - y) - 15

= 15(2x - y)² - 25(2x - y) + 9(2x - y) - 15

= (15(2x - y)² - 25(2x - y)) + (9(2x - y) - 15)

= 5(2x - y)(3(2x - y) - 5) + 3(3(2x - y) - 5)

= 10x - 5y(6x - 3y - 5) + 3(6x - 3y - 5)

= (6x - 3y - 5)( 10x - 5y + 3)

Hence, 15(2x - y)² - 16(2x - y) - 15 = (6x - 3y - 5)( 10x - 5y + 3)

301² x 300 - 300³

= 300 x (301² - 300²)

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

= 300 x (301 - 300) x (301 + 300)

= 300 x 1 x 601

= 1,80,300

Hence, 301² x 300 - 300³ = 1,80,300

$$\begin{aligned}\frac{(5z^2 - 80)}{(z - 4)}\\[1em] = \frac{5(z^2 - 16)}{z - 4}\end{aligned}$$

Using the formula

[∵ (x² - y²) = (x + y)(x - y)]

$$\begin{aligned}\frac{5(z^2 - 4^2)}{(z - 4)}\\[1em] = \frac{5(z - 4)(z + 4)}{(z - 4)}\\[1em] = \frac{5\cancel{(z - 4)}(z + 4)}{\cancel{(z - 4)}}\\[1em] = 5(z + 4)\end{aligned}$$

Hence, (5z² - 80) ÷ (z - 4) = 5(z + 4)

$$\begin{aligned}\frac{10y(6y + 21)}{(2y + 7)}\\[1em] \frac{10y \times 3(2y + 7)}{(2y + 7)}\\[1em] = \frac{30y(2y + 7)}{2y + 7}\\[1em] = \frac{30y\cancel{(2y + 7)}}{\cancel{(2y + 7)}}\\[1em] = 30y\end{aligned}$$

Hence, 10y(6y + 21) ÷ (2y + 7) = 30y

$$\begin{aligned}\frac{(a^2 - 14a - 32)}{(a + 2)}\\[1em] = \frac{(a^2 - (16 - 2)a - 32)}{(a + 2)}\\[1em] = \frac{(a^2 - 16a + 2a - 32)}{(a + 2)}\\[1em] = \frac{((a^2 - 16a) + (2a - 32))}{(a + 2)}\\[1em] = \frac{(a(a - 16) + 2(a - 16))}{(a + 2)}\\[1em] = \frac{((a - 16)(a + 2))}{(a + 2)}\\[1em] = \frac{(a - 16)\cancel{(a + 2)}}{\cancel{(a + 2)}}\\[1em] = (a - 16)\end{aligned}$$

Hence, (a² - 14a - 32) ÷ (a + 2) = (a - 16)

$$\begin{aligned}\frac{39x^3 (50x^2 - 98)}{26x^2(5x + 7)}\\[1em] = \frac{3x (50x^2 - 98)}{2(5x + 7)}\\[1em] = \frac{3x \times 2(25x^2 - 49)}{2(5x + 7)}\\[1em] = \frac{6x((5x)^2 - (7)^2)}{2(5x + 7)}\\[1em] = \frac{3x(5x - 7)(5x + 7)}{(5x + 7)}\\[1em] = \frac{3x(5x - 7)\cancel{(5x + 7)}}{\cancel{(5x + 7)}}\\[1em] = 3x(5x - 7)\end{aligned}$$

Hence, 39x³(50x² - 98) ÷ 26x²(5x + 7) = 3x(5x - 7)