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.

Yes, that is what you need to show.
Hint: Note that $[X^3 + 1] = [X]$ (using brackets to denote residue classes), and also $$[1] = [2X^3 + X] = [X][2X^2 + 1] = [X^3 + 1][2X^2 + 1].$$ Therefore...
