Factorise $a^{2} b^{2} - a^{3} b^{3} + 2 a b$

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Solution

$a^{2} b^{2} - a^{3} b^{3} + 2 a b = a^{2} b^{2} - 2 a^{3} b^{3} + a^{3} b^{3} + 2 a b = a^{2} b^{2} + a^{3} b^{3} - 2 a^{3} b^{3} + 2 a b = a^{2} b^{2} [1 + a b] - 2 a b [a^{2} b^{2} - 1] = a^{2} b^{2} [1 + a b] - 2 a b [a b + 1] [a b - 1] = (1 + a b) [a^{2} b^{2} - 2 a b (a b - 1)] = a b (1 + a b) [a b - 2 (a b - 1)] = a b (1 + a b) [a b - 2 a b + 2] = a b (1 + a b) (2 - a b)$

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