Sometimes with cubics you can find one zero just by inspection and then the other two by synthetic division. Not so in this case. Is this problem in a section about approximating roots (i.e, Newton's method)?
Ideally we would start with the rational roots theorem but here the only possible zeroes are -1 and 1 neither of which are zeroes.
Graphically is the best way to go here. the cubic equation is rather long and involved but:
If interested see Solving Cubic Equations
I would agree it would be simpler to use Newton's Method