Use the modulo version of the quadratic formula and Euler's criterion to decide if the following has a solution or not.
$2x^2+5x+8 \equiv 0\pmod{37}$
I'm not sure how I would use what was being asked of me to decide if this has a solution or not, but I have came to the conclusion that it does not, because I plugged in every possible answer $0-36$, and none of them were congruent to zero.
The way people have answered the question, isn't exactly the way I need. I have made some progress, but I'm not exactly sure how to finish:
$2x^2+5x+8 \equiv 0\pmod{37}$
$x^2+21x+4\equiv0\pmod{37}$
$x^2+58x+841\equiv837\pmod{37}$
$(x+29)^2\equiv23\pmod{37}$
I know that if $p$ is an odd prime and $p$ doesn't divide $a$, then $x^2\equiv a\pmod{p}$ has a solution or no solution depending on whether $a^\frac{p-1}{2}\equiv 1 \space\space \text{or}\space -1\pmod{p}$.
This is where I get stuck. Can someone help me from here? I have a feeling this problem has no solutions, so could someone also explain to me what to do if a problem like this did have solutions.