Factorise: ${(x^{2} + y^{2} - z^{2})}^{2} - 4 x^{2} y^{2}$

(x + y + z) (x + y - z) (x - y + z) (x + y + z)

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(x + y + z) (x + y - z) (x - y + z) (x - y - z)

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(x + y + z) (x + y - z) (x + y + z) (x - y - z)

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(x + y + z) (x + y - z) (x - y - z) (x - y - z)

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Solution

The correct option is B $(x + y + z) (x + y - z) (x - y + z) (x - y - z)$
${(x^{2} + y^{2} - z^{2})}^{2} - 4 x^{2} y^{2} = {(x^{2} + y^{2} - z^{2})}^{2} - {(2 x y)}^{2}$
$= {(x^{2} + y^{2} - z^{2} + 2 x y)} - (x^{2} + y^{2} - z^{2} + 2 x y) [∵ a^{2} - b^{2} (a - b) (a + b)]$
$= (x^{2} + 2 x y + y^{2} - z^{2}) (x^{2} - 2 x y + y^{2} - z^{2})$
$= [{(x^{2} + y^{2})}^{2} - z^{2}] [{(x^{2} - y^{2})}^{2} - z^{2}]$
$= (x + y + z) (x + y - z) (x - y + z) (x - y - z)$

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