How does $n+2\mid(n^2+n+1)(n^2+n+2)$ imply $n+2\mid12$? I'm trying to show that if $n+2\mid(n^2+n+1)(n^2+n+2)$, then $n+2 \mid 12$ for all $n\in \mathbb{N}$.

Hint
$$(n^2+n+1)(n^2+n+2)=n^4+2n^3+4n^2+3n+2$$
and
$$ n^4+2n^3+4n^2+3n+2=(n+2)(n^3+4n-5)+12$$
