Question

12(x^2+7x)^2-8(x^2+7x)(2x-1)-15(2x-1)^2
Factorise

Answer 1

Given Equation is 12(x^2 + 7x)^2 - 8(x^2 + 7x)(2x - 1) - 15(2x - 1)^2

= > 12(x^4 + 49x^2 + 14x^3) - 8(2x^3 + 13x^2 - 7x) - 15(4x^2 + 1 - 4x)

= > 12x^4 + 588x^2 + 168x^3 - 16x^3 - 104x^2 + 56x - 60x^2 + 60x - 15

= > 12x^4 + 152x^3 + 424x^2 + 116x - 15

= > 12x^4 + 104x^3 + 48x^3 - 10x^2 + 416x^2 + 18x^2  - 40x + 156x - 15

= > 2x^2(6x^2 + 52x - 5) + 8x(6x^2 + 52x - 5) + 3(6x^2 + 52x - 5)

= > (2x^2 + 8x + 3)(6x^2 + 52x - 5).


Hope this helps!

Answer 2

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FACTORISE : 12(x²+7x)² - (2x - 1) - 15(2x-1)²

✬Taking a = x² + 7x and b= 2x - 1, we have
12a² - 8ab - 15b²

Then, we can proceed of factories by middle term by splitting method.

The product of first and third term is

(12a²)(-15b²) = -180(ab)²
By observation, we can split the middle term -8ab into -18ab and 10ab whose product is the same as that of frist and third term.

∴ 12a² - 8ab - 15b²
= 12 a² - 18ab + 10ab - 15b²
= 6a(2a - 3b) + 5b(2a + 3b)
= (2a - 3b)(6a + 5b)

substituting a = x² + 7x and b = 2x -1,
we have,

→(2a - 3b)(6a + 5b)

= [2(x² + 7x)- 3(2x - 1)][6(x² + 7x)+5(2x - 1)]

= (2x² + 14x - 6x + 3)(6x² + 42x + 10x - 5)

= (2x² + 8x + 3)(6x² + 52x -5)

∴12(x² + 7x)²-8(x² + 7x)(2x - 1)- 15(2x -1)²

= (2x² + 8x +3)(6x² + 52x -5)✔
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