Solving Polynomial Equations

The passenger section of a train has width 2x-7, length 2x+3, and height x-2, with all dimensions in metres. Solve a polynomial equation to determine the dimensions of the section of the train if the volume is 117 m^3.

I tried this and I got the expanded equation 4x^3-16x^2-5x-75. When I tried solving it, it was unfactorable, but the answer at the back has whole number answers.
Can somebody help me please?

Agreed with
I think there is a real, integer root. Test the factors of 75: x=1, x=3, x=5, etc until you find it.