there is a "general formula" for polynomials of degree 2,3, and 4. be advised that this general formula is quite complex, and produces very nasty-looking solutions, in many cases. for polynomials of degree 5 or higher there is no "general formula" (although for "some" polynomials, we can still factor them easily).

your polynomial has integer coefficients. so the first thing one might check for is rational roots. these will be of the form ąp/q where q is an integer factor of 12, and p is an integer factor of 36.

that still gives a lot to check for, p might be 1,2,3,4,6,9,12,18 or 36, and q might be 1,2,3,4,6 or 12.

i come up with 1,2,3,4,6,9,12,18,36,1/2,1/3,1/4,1/6,1/12,2/3,3/4,3/2,9/2,9/4 and their negatives as rational roots to check. that 38 "easy checks".

entering this polynomial at Wolfram|Alpha yields: 12x^4 - 9x^3 - 58x^2 + 4x + 36 - Wolfram|Alpha

which indicates that the 4 roots of your polynomial are not rational. explaining how to get the "general solution" of a quartic is rather an advanced topic, if you feel up to it, see here:

Quartic function - Wikipedia, the free encyclopedia