Estimate of a sum of Legendre symbols of polynomials of even degree

Let $n\ge4$ be even, $p>\frac{n^2-2n}2$ be simple odd, and $f(x)=a_0+a_1x+\dots+a_nx^n$ be a polynomial with integral coefficients that are not quadratic over the residue field modulo $p$, $(a_n,p)=1$. The following inequality is proved:
$$ \biggl|\sum_{x=1}^p\biggl(\frac{f(x)}p\biggr)\biggr|\le(n-2)\sqrt{p+1-\frac{n(n-4)}4}+1. $$