Solving for x in a quadratic congruence modulo problem

So our class was given this to look over before our last class before the break:

$\displaystyle 5x^2 + 5x + 22 \equiv 0mod11$

I got a value that works, $\displaystyle x=11$

but not sure if this is the only answer and I went about it probably not in the correct fashion.

What I did was just notice that $\displaystyle 22 \equiv 0mod11$, then I though that if $\displaystyle x=11$, then $\displaystyle 55 \equiv 0mod11$, and it turns out that $\displaystyle 605 \equiv 0mod11$ as well.

I doubt you can just eliminate terms like this to make it work.

All the examples I've read here don't deal with anything more than say $\displaystyle 3x \equiv 1mod4$ or whatever. So I wasn't sure how to approach when a whole quadratic equation was on the LHS.

Also, the quadratic formula doesn't give any real solutions, so my original theory of using that doesn't work.

Any help would be appreciated, Thanks.