irreducibility of integer polynomials over $\mathbb{Q}$

Question

I need some help/hints on determining the irreducibility of the two functions:

1. $g(x)=x^p-p^2mx+(p-1)$, $p,m\in \mathbb{Z}$, $p$ a prime
2. $h(x)=x^4+4x^3+4x^2+4x+5$

I've tried using Eisenstein's criterion for the first question, but I can't seem to find a prime that would actually be useful. I've tried proving that it is irreducible in $\mathbb{Z}_p$, but that didn't work either.

For question 2, my thoughts are to first consider it in the complex plane, and go forth from there. But I have no idea how to make that work.

Answer 1

$g(x+1)=(x+1)^p-mp^2(x+1)+(p-1)=\displaystyle \sum_{k=0}^p\binom{p}{k}x^k-p^2m(x+1)+(p-1)$ the constant term of this is $p-p^2m=p(1-pm)$, as $1-pm \equiv 1 \pmod{p}$, this is not divisible by $p^2$. Hence $g(x+1)$ is Eisenstein with respect to $p$.

$h(x-1)=(x-1)^4+4(x-1)^3+4(x-1)^2+4(x-1)+5$ simplifies to $x^4-2x^2+4x+2$ which is $2$-Eisenstein.