Factorize the following expression:

$4 (x y + 1)^{2} - 9 (x - 1)^{2}$

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Solution

Given: $4 (x y + 1)^{2} - 9 (x - 1)^{2}$

$∴ 4 (x y + 1)^{2} - 9 (x - 1)^{2}$

$= [2 (x y + 1)]^{2} - [3 (x - 1)]^{2}$

Using the formula, $(a^{2} - b^{2}) = (a - b) (a + b)$

$= [2 (x y + 1) + 3 (x - 1)] [2 (x y + 1) - 3 (x - 1)]$

$= (2 x y + 2 + 3 x - 3) (2 x y + 2 - 3 x + 3)$

$= (2 x y + 3 x - 1) (2 x y - 3 x + 5)$

Hence, $4 (x y + 1)^{2} - 9 (x - 1)^{2} = (2 x y + 3 x - 1) (2 x y - 3 x + 5)$ .

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