Yeah it looks like a basic, really elementary question, but i'm having hard time with it.
First i tried to show that it's divisible by 9 $$(n-1)^3n^3(n+1)^3 = ((n+1)(n-1))^3n^3 = (n^2-1)^3n^3 = (n^3-n)^3$$ and using eulers theorem we know that $$[n^{\varphi(9)} \equiv 1 (mod \ 9)] = [n^6 \equiv 1 (mod \ 9)]$$ My doubt : can we do that? Cause $n$ and $9$ have to be coprime. Is it right direction? I'd love some help on this, cause i never did tasks which asks for proving divisiblity of some polynomial. Cheers!