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.

$$
(ax^2 + bx + c)(2x^2 + 2) + (dx + e)(x^3 + 2x + 2) = 1\\
(2a + d)x^4 + (2b + e)x^3 + (2a + 2c + 2d)x^2 + (2b + 2d + 2e)x + (2c + 2e) = 1\\
$$ 
$$\left[ \begin{array}{ccccc} 
2 & 0 & 0 & 1 & 0 \\ 
0 & 2 & 0 & 0 & 1 \\
2 & 0 & 2 & 2 & 0 \\
0 & 2 & 0 & 2 & 2 \\
0 & 0 & 2 & 0 & 2 \\
\end{array} \right] 
\left[ \begin{array}{c}
a \\ b \\ c \\ d \\ e \\ \end{array} \right]
=
\left[ \begin{array}{c}
0 \\ 0 \\ 0 \\ 0 \\ 1 \\ \end{array} \right]
$$
Solve for $a$, $b$, and $c$ and you find the inverse.
