Statement
The difference of two cubes can be factored as follows.
$$ a^3 - b^3 = (a - b)(a^2 + ab + b^2) $$
Proof
Take $ (a - b)(a^2 + ab + b^2) $ and expand the brackets. Then cancel like terms.
$$ (a - b)(a^2 + ab + b^2) = $$
$$ a^3 + a^2b + ab^2 - a^2b - ab^2 - b^3 = $$
$$ a^3 - b^3 $$