Resolve $a^{3} - b^{3} + 1 + 3 a b$ into factors.

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Solution

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

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