Find the prime ideals of $\mathbb Z[i][x]/(1+i, x^2+2).$
We know that $(1+i)(1-i)=2$ and so $\mathbb Z[i][x]/(1+i,x^2+2) = \mathbb Z[i][x]/(1+i,x^2).$ Now, any prime ideal that contains $(1+i,x^2)$ must contain $x;$ thus, the prime ideal contains the ideal $(1+i,x).$ However, $\mathbb Z[i][x]/(1+i,x) \cong \mathbb Z / 2 \mathbb Z,$ and so $(1+i,x)$ is a maximal ideal, i.e., it is the only prime ideal. Hence, there is only one prime ideal: it is of the form $(1+i,x)/(1+i,x^2).$ Is this correct? Is there a neater way to denote this ideal?
That is right. Note that $\mathbb Z[i]/(1+i)$ is just $\mathbb F_2$, so you really have $\mathbb F_2 [x]/(x^2),$ and your ideal is the one generated by the image of $x$ in that ring.