Unit of a quotient ring. I'm having trouble showing whether this is true or false and why:
The class $X^3+1+(X^3+2X+1)$ is a unit in the quotient ring $\Bbb Z_3[X]/(X^3+2X+1)$
I think I need to show that it has a multiplicative inverse but not sure if that's right or how.
Any help would be greatly appreciated.

$x^3+1 = -2x$ mod $x^3+2x+1$ and $1 = -x^3-2x$ mod $x^3+2x+1$. Since $\mathbb{Z}_3$ is the ground ring we have $-2x = x$ and $1 = 2x^3+x$ in $\mathbb{Z}_3[x]/(x^3+2x+1)$. Therefore $1 = x(2x^2+x)$ in $\mathbb{Z}_3[x]/(x^3+2x+1)$. Hence $(x^3+1)^{-1}= 2x^2+x$.
