Factorize $4 x^{4} - x^{2} - 12 x - 36$

(2 x + 3) (x + 2) (2 x^{2} + x + 6)

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(2 x - 3) (x - 2) (2 x^{2} + x + 6)

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= (2 x + 3) (x - 2) (2 x^{2} + x + 6)

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(2 x + 3) (x - 2) (2 x^{2} - x - 6)

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Solution

The correct option is C $= (2 x + 3) (x - 2) (2 x^{2} + x + 6)$

$4 x^{4} - x^{2} - 12 x - 36$
$= (2 x^{2})^{2} - (x^{2} + 12 x + 36)$
$= (2 x^{2})^{2} - (x^{2} + 6 x + 6 x + 36)$
We have $a^{2} + b^{2} + 2 a b = (a + b)^{2}$ and $a^{2} - b^{2} = (a + b) (a - b)$
$= (2 x^{2})^{2} - (x + 6)^{2}$
$= (2 x^{2} - x - 6) (2 x^{2} + x + 6)$
$= (2 x + 3) (x - 2) (2 x^{2} + x + 6)$

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