An irreducible polynomial in Z[x]

Question

Let $p$ be a prime and let $n$ be an integer greater than $4$. Prove that if $a$ is an integer that is not divisible by $p$, then the polynomial $$f(x)=ax^n-px^2+px+p^2$$is irreducible over $\mathbb{Z}[x]$.

Note. I have proven that $f(x)=(bx+p)g(x)$, where $g(x) \in \mathbb{Z}[x]$ and $b \in \mathbb{Z}$. I have also proven that $|b|>1$.

Answer 1

There is a counterexample :

Let $p=2$ , $n=5$ , $a=1715$

Then $$f(x)=1715x^5-2x^2+2x+4$$ is divisible by $$7x+2$$