factorise :
$12 (x^{2} + 7 x)^{2} - 8 (x^{2} + 7 x) (2 x - 1) - 15 (2 x - 1)^{2}$

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Solution

$12 (x^{2} + 7 x)^{2} - 8 (x^{2} + 7 x) (2 x - 1) - 15 (2 x - 1)^{2}$
$= 12 (x^{2} + 7 x)^{2} - 18 (x^{2} + 7 x) (2 x - 1) + 10 (x^{2} + 7 x) (2 x - 1)$
$- 15 (2 x - 1)^{2}$
$= 6 (x^{2} + 7 x) [2 (x^{2} + 7 x) - 3 (2 x - 1)] +$
$5 (2 x - 1) [2 (x^{2} + 7 x) + 3 (2 x - 1)]$
$= [2 (x^{2} + 7 x) - 3 (2 x - 1)] [6 (x^{2} + 7 x) + 5 (2 x - 1)]$
$= [2 x^{2} + 14 x - 6 x + 3] [6 x^{2} + 42 x + 10 x - 5]$
$= (2 x^{2} + 8 x + 3) (6 x^{2} + 52 x - 5)$
$∴ 12 (x^{2} + 7 x^{2})^{2} - 8 (x^{2} + 7 x) (2 x - 1) - 15 (2 x - 1)^{2}$
$= [2 (x^{2} + 7 x) - 3 (2 x - 1)] [6 (x^{2} + 7 x) + 5 (2 x - 1)]$
$= (2 x^{2} + 8 x + 3) (6 x^{2} + 52 x - 5)$

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