$\left(x^{3}+x^{2}+1\right)\left(x^{2}+x+1\right)=x^{5}+x+1 \equiv x^{2}+1 \bmod x^{4}+x+1=0101$
I have this equation and I don't get how to get to the answers?
Is there a fast method to get to the result?
And I don't get the modulo part how do I come up to this myself?
Could someone explain me this simple and maybe a general way to approach things like this fast?
I'm assuming you're taking coefficients modulo two; if not, you can clarify what you mean and I'll change my answer.
Multiply it out: $$ (x^3+x^2+1)(x^2+x+1) = 1 + x + 2 x^2 + 2 x^3 + 2 x^4 + x^5 $$Take coefficients mod two: $$ \to 1 + x + x^5 $$Now compute modulo $x^4+x+1$. This means $x^4+x+1=0$, or $x^4=x+1$, or $x^5=x^2+x$: $$ \to 1+x+(x^2+x)\to 1+x^2 $$