Show that as a polynomial in $\mathbb{Q}[x]$, $P(x)$ is irreducible. Let $P(x)$ be a polynomial in $\mathbb{Z}[x]$ of degree 5 such that $P(1)=3$ and $P=(x-1)^5\bmod 3$. Show that as a polynomial in $\mathbb{Q}[x]$, $P(x)$ is irreducible.

So here I find that $$P(x)=(x-1)^5+3=x^5-5x^4+10x^3-10x^2+5x+2.$$ How can I show this irreducible? Can I use Eisenstein's Criterion? How would I do that? For instance, if my prime is 5 then it divides 4 of the 6 terms but what about those other 2?
$\!\bmod 3\!:\ p \equiv (x\!-\!1)^5\ $ so Eisenstein applies to $ f(x) := p(x\!+\!1) \equiv x^5\,$ by $\,3^2\nmid f(0)\! =\! p(1)\!=\! 3$

Remark $ $ This is the standard shifted Eisenstein criterion, e.g. see here for motivation.
It suffices to show it is irreducible over $\Bbb Z$, by Gauß' lemma.  Go mod $3$ and use Eisenstein.