The expression $(x^{2} + 4 y^{2} - 9 z^{2})^{2} - 16 x^{2} y^{2}$ , is equivalent to

(x - 2 y + 3 z) (2 x - 2 y - 3 z) (x - 2 y + 3 z) (x + 2 y - 3 z)

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(x + 2 y + 3 z) (x + 2 y + 3 z) (x - 2 y + 3 z) (x + 2 y - 3 z)

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(x - 2 y + 3 z) (x + 2 y - 3 z) (x - 2 y + 3 z) (x - 2 y - 3 z)

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(x + 2 y + 3 z) (x + 2 y - 3 z) (x - 2 y + 3 z) (x - 2 y - 3 z)

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Solution

The correct option is D $(x + 2 y + 3 z) (x + 2 y - 3 z) (x - 2 y + 3 z) (x - 2 y - 3 z)$
$(x^{2} + 4 y^{2} - 9 z^{2})^{2} - 16 x^{2} y^{2}$
$= (x^{2} + 4 y^{2} - 9 z^{2})^{2} - (4 x y)^{2}$
$= (x^{2} + 4 y^{2} - 9 z^{2} + 4 x y) (x^{2} + 4 y^{2} - 9 z^{2} - 4 x y)$
$= (x^{2} + 4 x y + 4 y^{2} - (3 z)^{2}) (x^{2} - 4 x y + 4 y^{2} - (3 z)^{2})$
$= (x^{2} + 2 (x) (2 y) + 4 y^{2} - (3 z)^{2}) (x^{2} - 2 (x) (2 y) + 4 y^{2} - (3 z)^{2})$
$= ((x + 2 y)^{2} - (3 z)^{2}) ((x - 2 y)^{2} - (3 z)^{2})$
$= (x + 2 y + 3 z) (x + 2 y - 3 z) (x - 2 y + 3 z) (x - 2 y - 3 z)$

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