Factorize :
$(a^{2} - b^{2}) (c^{2} - d^{2}) - 4 a b c d$

(a c - b d + a d + b c) (a c - b d - a d - b c)

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(a - d + a d + c) (b d - a d - b c)

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(a c - b d + a d - b c) (a c - b d - a d - b c)

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(a c - b d + a d + b) (a c - b d - b c)

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Solution

The correct option is A $(a c - b d + a d + b c) (a c - b d - a d - b c)$
Given, $(a^{2} - b^{2}) (c^{2} - d^{2}) - 4 a b c d$
$= a^{2} c^{2} - a^{2} d^{2} - b^{2} c^{2} + b^{2} d^{2} - 4 a b c d$
$= (a^{2} c^{2} + b^{2} d^{2} - 2 a b c d) - (a^{2} d^{2} + b^{2} c^{2} + 2 a b c d)$
Using,
$(a - b)^{2} = a^{2} - 2 a b + b^{2}$ and $(a + b)^{2} = a^{2} + 2 a b + b^{2}$
$\Rightarrow$ $(a c - b d)^{2} - (a d + b c)^{2}$

Using,
$(a^{2} - b^{2}) = (a + b) (a - b)$
$\Rightarrow$ $(a c - b d + a d + b c) (a c - b d - a d - b c)$

Suggest Corrections

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