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.

This is in no way a standard solution, but I just want to include it for fun. 
Note that $x^3=-(2x^2+2)$, we will find $x^{-3}$. Now
$$x^3-x^2=-2=1$$
implies 
$$x(x^2-x)=x^2(x-1)=1.$$
Thus 
$$x^{-1}=x(x-1)\text{ and }x^{-2}=x-1.$$
Multiplying the two we get
$$x^{-3} = x(x-1)^2=x(x^2+x+1)=x^3+x^2+x.$$
Using $x^3=x^2+1$ on the RHS, we see that
$$x^{-3}=2x^2+x+1.$$
Thus
$$(2x^2+2)^{-1} = -x^{-3} = -2x^2-x-1 = x^2+2x+2.$$
