Then I would use the "rational root theorem": any rational roots of this equation must be a fraction with numerator a factor of 3 and denominator a factor of 4. That is, the numerator must be one of , or and the denominator must be one of , , or . There are 12 possible combinations. Try each of those. If you find a rational root among those, then you can reduce the equation to a quadratic to find the other roots.
If there is no rational root, then you will just have to use Cardano's formula, which is terribly messy.