determining if quotient ring of polynomials over a finite field is a field or not

Question

I am stuck with this question:

"Determine if $GF(2011^2) [x] /<x^4-6x+12>$ is a field or not."

I know that since this polynomial ring is defined over a field, I only have to determine if $x^4-6x+12$ is irreducible or not. But I have no idea how to do this.

Professor told me to check what happens if I assume this polynomial to be reducible. But I'm still in a mist.

Thank you in advance for help.