Factorise the following: $a^{2} b^{2} + c^{2} d^{2} - a^{2} c^{2} - b^{2} d^{2}$

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Solution

Given that: $a^{2} b^{2} + c^{2} d^{2} - a^{2} c^{2} - b^{2} d^{2}$
$= (a^{2} b^{2} - a^{2} c^{2}) + (c^{2} d^{2} - b^{2} d^{2})$ , (group pair of terms which have common)
$= a^{2} (b^{2} - c^{2}) - d^{2} (b^{2} - c^{2})$ (factor each binomials)
$= (b^{2} - c^{2}) (a^{2} - d^{2})$ , (factor out common factor $(b^{2} - c^{2})$ )
$= (b + c) (b - c) (a + d) (a - d)$ ,------(since $x^{2} - y^{2} = (x + y) (x - y)$ )

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