By trial and error we can see x=2 is a solution, so we can divide by this factor using synthetic division to get a quadratic. (The constant -8 suggests that at least one factor contains the constant 1, 2, 4 or 8.)
2 | 3 2 -12 -8
3 6 16 8
3 8 4 | 0 = 3x²+8x+4 = (3x+2)(x+2).
So the complete factorisation is (x-2)(3x+2)(x+2). Therefore x=-2, -2/3, 2.