This is the same as: 2x^5-7x^4-7x^3+24x^2+10x-12=0
Inspection reveals that x=-1 is a root, also that x=2 is a root. That gives us two factors: x+1 and x-2. The product of these gives us a constant -2. The remaining factors must therefore have a product of -12/-2=6. First divide by the known factors using synthetic division: 2x^3-5x^2-8x+6=0. Divide through by 2: x^3-5x^2/2-4x+3=0. Neither x=3 or x=-3 are roots. Try x=3/2 and -3/2. We find x=-3/2 is a zero. So we now divide by the factor x+3/2 or the root -3/2: x^2-4x+2=0; x^2-4x+4-4+2=0; (x-2)^2=2; x-2=+sqrt(2) and x=2+sqrt(2)=3.4142 and 0.5858.
We have five roots: -1.5, -1, 2, 2-sqrt(2), 2+sqrt(2).