I need some help with this exersise because it mixes quotients rings, concruences, ideals, polynomials and it mess me up. I don't even know how to start. Any help will be appreciated.
Let $R=\Bbb Z / 3 \Bbb Z$ and $I$ the ideal generated by $p(x)=x^2+[2]_3x+[1]_3$ in $R[x]$.
(a) Is $R[x]/I$ integral domain? If so, prove it.
(b) Is $([2]_3x+[1]_3)+I$ a unit in $R[x]/I$? If so, find the inverse of $([2]_3x+[1]_3)+I$ using the Euclidean Algorithm.
(c) Let $S:=R[x]/I$. Calculate $S/((x+[1]_3)+I)$ and say whether $(x+[1]_3)+I$ is prime or not in $S.$
What I've done so far:
(a) As $p(x)=(x+[1]_3)^2$ is not prime and R[x] is a PID then the ideal generated by p(x) is not prime so that $R[x]/I$ is not an integral domain. (Thanks for the hints in the comments.)
(b) $gcd(x^2+[2]_3x+[1]_3, \; [2]_3x+[1]_3)=1$ so I get $x^2+[2]_3x+[1]_3=([2]_3x)([2]_3x+[1]_3)$. Thus, $$0+I=(([2]_3x)([2]_3x+[1]_3)+1)+I,$$ and $$1+I=(([2]_3x)([2]_3x+[1]_3))+I=(([2]_3x)+I)\cdot (([2]_3x+[1]_3)+I),$$ so the multiplicative inverse of $([2]_3x+[1]_3)+I$ is $[2]_3x+I$. $\color{red}{Am\; I \; wrong?}$
(c) I really don't know what to do.