In $\mathbb{Z}_2[x]$, the ideal $\langle x^2+x+1\rangle$ is maximal 
I want to show that in $\mathbb{Z}_2[x]$, the ideal $\langle x^2+x+1\rangle$ is maximal ideal. 

I have listed out the table and I can see that the quotient ring is isomorphic to $\mathbb{Z}_2\oplus \mathbb{Z}_2$, but I can't find a ring homomorphism $f:\mathbb{Z}_2[x]\rightarrow \mathbb{Z}_2\oplus \mathbb{Z}_2$ such that the kernel of this homomorphism is exactly the ideal $\langle x^2+x+1\rangle$. 
Also, can anyone give me any general suggestion of finding such homomorphisms?

This ideal is maximal since any element $ax+b+ < x^2+x+1>$ has an inverse. To find the inverse see that $(ax+b)(cx+d)=(acx^2+(ad+bc)x+bd=(ad+bc+1)x+(bd+1)$ modulo $<x^2+x+1>$. So if $a,b$ both are not zero we can solve for $c,d$ from the equations: $ad+bc+1=0, bd+1=1$. 
You got the quotient ring wrong: for example $x^2= x+1$ modulo $x^2+x+1$ but all the elements in $\mathbb{Z}_2\times \mathbb{Z}_2$ are idempotents.
