Eisenstein's criterion

Eisenstein. $P$ has integer coefficients. For $p=2$:
- $2\mid$ all non-leading coefficients: $2,2,2,2$.
- $2\nmid$ leading coefficient $1$.
- $4\nmid$ constant term $2$.

By Eisenstein's criterion, $P$ is irreducible over $\mathbb{Q}$.

Conclusion. $P$ is irreducible over $\mathbb{Q}$.