Factorise:

$(a^{2} - 1) (b^{2} - 1) + 4 a b$

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Solution

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

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