Which of the following polynomials below is identical to the polynomial $(x^{2} - 1) (x^{2} - 4)$ ?

$x^{4} + 4$

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$x^{2} + 5 x + 4$

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$(x^{2} + 1) (x^{2} + 4)$

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$(x^{2} + x - 2) (x^{2} - x - 2)$

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Solution

The correct option is D
$(x^{2} + x - 2) (x^{2} - x - 2)$

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

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