<s>[INST] Find the inverse of a polynomial in a quotient ring Consider $\mathbb Z_5[x]/I$ with $I$ as ideal generated by $b=x^3+3x+2$. If $(x+2) + I$ is element of $\mathbb Z_5[x]/I$ that has an inverse. Find the inverse of $(x+2) + I$.
I stuck to get the inverse because the gcd of that is not $1$.
 [/INST] Hint: $$(x+2)(ax^2+bx+c) + I =ax^3+(2a+b)x^2+(2b+c)x+2c+I\\ = (2a+b)x^2 +(2b+c-3a)x+2(c-a) + I$$
Now solve
\begin{align}
2a+b &=0\\
2b+c-3a &= 0\\
2(c-a) &= 1 
\end{align}
in $\Bbb Z_5$
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