I've always been able to compute the Galois groups of polynomials of degree $\leq 4$, but I have trouble at higher degrees. I can factor quadratics and cubics, and get the solutions from there, but when I run into a polynomial I don't know how to factor, this is a problem. Is there a canonical method to calculating these Galois groups?

In particular, how do I go about calculating the Galois group of $x^5-12x+2$? I don't know how to factor that polynomial, so I don't really know how any of the roots work.