Is $X^4+X^3+X+1$ reducible?

Question

The polynomial $X^4+X^3+X+1$ over $\mathbb{k}[x]$ does not seem to be reducible, but I cannot find a way to prove it. Eisenstein criteria doesn't apply here, I think.

I tried some alternative form, thinking something like $(x^2+1)^2+x^3+1-2x^2$ may make it clear but it doesn't.

Answer 1

No, because: $\ x^{4}+x^{3}+x+1=x^{3}(x+1)+(x+1)=(x+1)(x^{3}+1)$.