Factorise: $12 {(a^{2} + 7 a)}^{2} - 8 (a^{2} + 7 a) (2 a - 1) - 15 {(2 a - 1)}^{2}$

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Solution

$12 (a^{2} + 7 a)^{2} - 8 (a^{2} + 7 a) (2 a - 1) - 15 (2 a - 1)^{2}$

Let $(a^{2} + 7 a) = x$ and $(2 a - 1) = y$

Then the previous equation becomes:

$12 x^{2} - 8 x y - 15 y^{2}$

$= 12 x^{2} - 18 x y + 10 x y - 15 y^{2}$

$= (12 x^{2} - 18 x y) + (10 x y - 15 y^{2})$

$= 6 x (2 x - 3 y) + 5 y (2 x - 3 y)$

$= (6 x + 5 y) (2 x - 3 y)$

Substituting the values of x and y:

$= 6 (a^{2} + 7 a) + 5 (2 a - 1) 2 (a^{2} + 7 a) - 3 (2 a - 1)$

$= (12 a^{2} + 42 a + 10 a - 5) (2 a^{2} + 14 a - 6 a + 3)$

$(12 a^{2} + 52 a - 5) (2 a^{2} + 8 a + 3)$

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