Could somebody please give me some help on this subject.

I thought in Z_{2}[x] the polynomial x^{5}+x^{4}+1 can be factored as (x^{3}+x+1)(x^{2}+x+1)?

I am confused as the user JavaMan on this website abstract algebra - Find all irreducible monic polynomials in $\mathbb{Z}/(2)[x]$ with degree equal or less than 5 - Mathematics Stack Exchange lists x^{5}+x^{4}+1 as a irreducible polynomial, and expands (x^{3}+x+1)(x^{2}+x+1) to x^{5}+x^{4}+x^{2}+x+1.

I am doing this correctly? If not please explain what I am doing wrong.