Find all positive integers $n$ such that $n^5 - 5n^3 + 5n + 1 | n!$
I know that $ n^5-5n^3+5n+1=(n+1)(n^4-n^3-4n^2+4n+1)$, but I have no idea where to go from here.
This was from a local contest.
If there are an infinite amount, I would like to know like a "general" solution.
Possibly, the explicit despiprion of such $n$ is beyond our possibilities. But for infiniteness of such $n$ user Math1Zzang on mathlinks proved the next generalization of your problem (post #3 here: https://artofproblemsolving.com/community/c6h1793837p11880596):
For every positive integer $m$, there exists polynomial $P_{m}(x)$ such that $P_{m}\left(x+\frac{1}{x}\right)=x^{m}+\frac{1}{x^m}$. Then there exists infinitely many positive integers $n$ for which $P_{m}(n)+1|n!$