Finding indempotents in a quotient ring 
I am trying to find the nontrivial indempotents in the ring $\mathbb{Z_3}[x]/(x^2+x+1)$. 

We can clearly see that $0,1$ are indempotents. I want to prove they are the only ones. Thus I am wondering if there is a way other than just brute force to show this. Currently I am down to 9 possibilities but would like to avoid checking each one individually

Any element of the ring can be written (uniquely) in the form $ax+b$.  If such an element is an idempotent, then $(ax+b)^2=ax+b$.  Expand this, use the relation $x^2 = - x -1$, and equate coefficients to get a system of two equations for $a$ and $b$.  If that system has a solution in $\mathbb{Z}_3$ then you have found an idempotent; if it has no solution, then you have shown no idempotent exists.
