Whoops, I didn't remember you were looking for real solutions.
I don't see why you set [itex] x^3-7x^2+14x-8=12 [/itex], since it isn't said that this polynomial is the volume. I'd suggest factorising [itex] (x^3-7x^2+14x-8) [/itex] instead, and reading off the dimensions of the box from the factorization. I think that is what is asked for.
edit: Nope that doesn't make sense either. You were of course correct.