What a Proof Is · Medium problem

$n^4 + 4 = (n^2 + 2n + 2)(n^2 - 2n + 2)$

$n^3 - n = (n - 1)n(n + 2)$

$(n + 1)^3 - n^3 = 3n^2 + 3n + 1$

$n(n + 1)(n + 2) + (n + 1) = (n + 1)^3$