On the largest prime divisor of polynomial and related problem

We denote P = (x)| P(n) n! for infinitely many n\. This article identifies some polynomials that belong to 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 ∈ N so that P⁺(f(n)) < n3/4+ε if $f(x)$ is cubic polynomial, P⁺(f(n)) < n if $f(x)$ is reducible quartic polynomial and P⁺(f(n)) < nε if $f(x)$ is Chebyshev polynomial.