IMO 2005 — Complete Shortlist Question 27

Let $ P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0 $, where $ a_0, \ldots, a_n $ are integers, $ a_n \gt 0 $, $ n \geq 2 $. Prove that there exists a positive integer $ m $ such that $ P(m!) $ is a composite number.