The inverse of $2x^2+2$ in $\mathbb{Z}_3[x]/( x^3+2x^2+2)$ What is the independent coefficient in the inverse of $2x^2+2$ in $\mathbb{Z}_3[x]/(x^3+2x^2+2)$ ?
I have been calculating some combinations, but I don't know how I can calculate the inverse.

It must take the form $ax^2+bx+c$.
Multiply this by $2x^2+2$, and you get a degree-four polynomial.
Since $x^3+2x+2=0$, it follows that $x^3=x+1$ and so $x^4=x(x^3)=x^2+x$.
So your degree-four polynomial becomes a quadratic.
This quadratic equals 1, or $0x^2+0x+1$.
Equate the three coefficients, and get three linear equations in $a,b,c$.
