If $p\in\mathbb{N}$ is a prime, is $x^n+px+p^2$ irreducible in $\mathbb{Z}[x]$?
I've proved that any non-unit factor in $\mathbb{Z}[x]$ must have degree at least 2.
Eisenstein's criterion doesn't hold, but perhaps we can make some minor change so that it does? (E.g. a linear substitution, as with cyclotomic polynomials??)
Or could we start from scratch, assuming $x^n+px+p^2 = (b_kx^k+...+b_0)(c_{n-k}x^{n-k}+...+c_0)$ and trying to get conditions on the $b$'s and $c$'s that give a contradiction. I've got that $b_0=\pm p$, $c_0=\pm p$, $b_1=0$, $c_1=\pm 1$, and $c_2=\pm b_2$, but can't see how this helps!
A hint given in the question says 'Consider powers of $p$ dividing coefficients.' Does this suggest some variant of Eisenstein, or an unusual way of implementing Eisenstein?
Thanks for any help with this!