You problably mean 2x3+3x2-12x-45=0 because the given equation has no rational roots and synthetic division would be unsuitable. If there is a rational root (making synthetic division feasible) then it's probably a factor of 45, which has 3 and 5 as its factors.
When x=3, we have 54+27-36-45=0, therefore x-3 is a factor and we can divide by this zero using synthetic division:
3 | 2 3 -12 | -45
2 6 27 | 45
2 9 15 | 0 = 2x2+9x+15 which has complex roots:
2x2+9x+15=0,
x2+4.5x=-7.5,
x2+4.5x+5.0625=-7.5+5.0625=-2.4375,
(x+2.25)2=-2.4375=-39/16.
x+2.25=±i√39/4=±1.56125i approx, x=-2.25+1.56125i, or -2.25-1.56125i.
So the roots are 3 and -2.25+1.56125i, and -2.25-1.56125i.