We look for rational zeroes first. These will be factors of 12: 1, 2, 3, 4, 6, 12. There could be plus or minus in front of these.
Try x=1: 1+2-17+2+12=0, so x-1 is a factor. We can divide by this factor using synthetic division:
x³+3x²-14x-12. We still have same factors, but x=1 no longer works. Try x=-1: -1+3+14-12=4. x=-1 doesn’t work either.
Try x=2: 8+12-28-12=-20 so x-2 is not a factor.
Try x=-2: -8+12+28-12=20 so x+2 is not a factor.
Try x=3: 27+27-42-12=0, so x-3 is a factor. We divide again: x²+6x+4 and complete the square:
x²+6x+9-9+4=0, (x+3)²-5=0=(x+3+√5)(x+3-√5)=0.
So the remaining roots are -3-√5 and -3+√5, which evaluate approximately to -5.2361, -0.7639.
The roots are 1, 3, -5.2361, -0.7639.