Factorize $(a + b)^{3} - 8$

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Solution

$(a + b)^{3} - 8$
$= (a + b)^{3} - 2^{3}$
Applying the formula $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$
$= (a + b - 2) [(a + b)^{2} + 2 (a + b) + 2^{2}]$
$= (a + b - 2) (a^{2} + 2 a b + b^{2} + 2 a + 2 b + 4)$

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