On the largest prime divisor of polynomial and related problem

We denote P = \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) ) < n^3{4+} if f (x) is cubic polynomial, P^+ (f (n) ) < n if f (x) is reducible quartic polynomial and P^+ (f (n) ) < n^ if f (x) is Chebyshev polynomial.