Factorize the equation ${(x^{2} + 4 y^{2} - 9 z^{2})}^{2} - 16 x^{2} y^{2}$

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

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

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

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(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 $(x + 2 y + 3 z) (x + 2 y - 3 z) (x - 2 y - 3 z)$
${(x^{2} + 4 y^{2} - 9 z^{2})}^{2} - 16 x^{2} y^{2}$
$= (x^{2} + 4 y^{2} - 9 z^{2})^{2} - (4 x y)^{2}$
Using,
$a^{2} - b^{2} = (a + b) (a - b)$
$= (x^{2} + 4 y^{2} - 9 z^{2} + 4 x y) (x^{2} + 4 y^{2} - 9 z^{2} - 4 x y)$
Using,
$(a + b)^{2} = a^{2} + 2 a b + b^{2}$ and $(a - b)^{2} = a^{2} - 2 a b + b^{2}$
$= [(x + 2 y)^{2} - 9 z^{2}] [(x - 2 y)^{2} - 9 z^{2}]$
$= (x + 2 y + 3 z) (x + 2 y - 3 z) (x - 2 y + 3 z) (x - 2 y - 3 z)$

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