Consider a polynomial of finite degree with integer coefficients P(N), does P(N) divide N! for infinitely many integer N?

This question is motivated by the special case where P(N) = N^2 + 1 that appeared in a math olympiad.

I was wondering if anyone can point me to references of this question.

Consider a polynomial $P(X)\in\mathbb Z[X]$. Is it true that $P(N)$ divides $N!$ for infinitely many integer $N$?

This question is motivated by the special case where $P(X) = X^2 + 1$ that appeared in a math olympiad.

I was wondering if anyone can point me to references of this question.