On the largest prime divisor of polynomial and related problem

We denote $\mathcal{P}$ = $\{P(x)\mathrm{\mid }$ $P(n)\mid n!$ for infinitely many $n\}$. This article identifies some polynomials that belong to $\mathcal{P}$. Additionally, we also denote $P^{+}(m)$ as the largest prime factor of $m$. Then, a consequence of this work shows that there are infinitely many $n\in \mathbb{N}$ so that $P^{+}(f(n))<n^{\frac{3}{4}+\epsilon }$ if $f(x)$ is cubic polynomial, $P^{+}(f(n))<n$ if $f(x)$ is reducible quartic polynomial and $P^{+}(f(n))<n^{\epsilon }$ if $f(x)$ is Chebyshev polynomial.