Factor the polynomial into irreducible factors in the field $\mathbb{Z} / 2\mathbb{Z_{[x]}}$

Question

I want to factor the $X^5 + X^3 + X^2+1$ into irreducible factors in the field $\mathbb{Z} / 2\mathbb{Z[x]}$.

I'm new in abstract-algebra, so I couldn't apply ideas from similar tasks to this one.

So, can someone explain it for me on this case? Looks like I have to brute-forse some roots and then compose their multiplies into $\mathbb{Z} / 2\mathbb{Z_{[x]}}$, but I'm not sure.

Answer 1

$$x^5+x^3+x^2+1=x^3(x^2+1)+x^2+1=(x^3+1)(x^2+1)=$$ $$=(x+1)(x^2-x+1)(x^2+2x+1)=(x+1)^3(x^2+x+1).$$ $x^2+x+1$ is irreducible because $1$ and $0$ are not roots of this polynomial.