Factorize: $(1 - x^{2}) (1 - y^{2}) + 4 x y$

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Solution

$(1 - x^{2}) (1 - y^{2}) + 4 x y$
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$= 1 - x^{2} - y^{2} + x^{2} y^{2} + 4 x y$
$= 1 - x^{2} - y^{2} + x^{2} y^{2} + 2 x y + 2 x y$
$= 1 + x^{2} y^{2} + 2 x y - (x^{2} + y^{2} - 2 x y)$
$= (1 + (x y))^{2} - (y - x)^{2} [a^{2} - b^{2}]$
$= (1 + x y + y - x) (1 + x y - y + x) a^{2} - b^{2} = (a + b) (a - b)$

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