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

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Solution

$1 + 2 a b - (a^{2} + b^{2}) = 1 + 2 a b - a^{2} - b^{2} = 1 - a^{2} + 2 a b - b^{2} = 1^{2} - (a^{2} - 2 a b + b^{2}) = 1^{2} - (a - b)^{2} s i n c e [a^{2} - 2 a b + b^{2} = (a - b)^{2}] = [1 + (a - b)] [1 - (a - b)] s i n c e [a^{2} - b^{2} = (a + b) (a - b)]$

$= (1 + a - b) (1 - a + b)$

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106

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