Prove, in $ \displaystyle \mathbb{Z} \left[ x \right]$, the irreducibility of
$ \displaystyle p + \sum_{k=1}^n x^k$
where $n>1$ and $p$ a prime number.
It is not too difficult to exploit the magnitude of the roots to create a contradiction under the assumption that the polynomial factors.
However, I am wondering if there are other means of proving the irreducibility of this polynomial, such as through Eisenstein's Criterion.