Division Algorithm and Polynomials.

Question

By the Division Algorithm, I know I can reduce every polynomial in $\mathbb{Z}_2[x]$ to a polynomial $ax+b$ mod $x^2+1$.
There are two possibilities for $a$ and two for $b$, a total of 4.
It follows that $\mathbb Z_{2}[x]/(x^{2}+1)$ is a field with 4 elements.
This is where I continually keep getting stuck. I know two of the elements should be 0,1 but I don't know how to find all of the elements.

Answer 1

$x, x+1,1,0$ are the solutions