Find all homomorphisms from a quotient polynomial ring $\mathbb{Z}[X] /(15X^2+10X-2)$ to $\mathbb{Z}_7$ I'm completely lost, what my problem is I don't get the gist of a quotient polynomial ring nor ANY homomorphisms between it and some $\mathbb{Z}_n$, much less ALL of them. 
I know there is something to be done with an ideal, but I really have no clue how to do it. I would be grateful for a full solution or at least anything that may help me understand these structures. 
EDIT:
Ok, so this particular polynomial was inreducible in $\mathbb{Z}_7$, but I still have no clue how to deal with other cases, where it is reducible, for instance $X^2+3X+3$. I really would like to know how to solve problems of this kind. 

Hint: a ring homomorphism $\Bbb Z[X]/(\cdots)\to\Bbb Z/7\Bbb Z$ will be determined by where $X$ is sent. It can't be sent just anywhere; it still has to satisfy $15X^2+10X-2=0$ (does this have roots in $\Bbb Z/7\Bbb Z$?).
