For f(x) to have at most one root on [-2,2], it must be strictly increasing or decreasing on that interval.
So let's examine f'(x) on the interval [-2,2].
Absolute maximum on [-2,2] is when x=±2, f'(x)=4-15=-11
Absolute minimum on [-2,2] is when x=0, f'(x)=-15.
Therefore f'(x) is negative on [-2,2], therefore strictly decreasing. Under these circumstances, f(x) can have at most one zero (root).