Judging by the plot, the polynomial has 2 roots (at -1 and 2), so its degree is at least 2. But if it were 2, we'd expect a parabolic shape, but that's not the case here. In fact, the root at -1 has even multiplicity, which means the degree of the polynomial should be at least 3. But again, you're probably familiar enough with the shape of a cubic function to know that the degree would be higher than that, so at minimum, I think your answer of 4 is correct.
We can use the tools of calculus to verify this. We know of two extrema (a max at -1 and a min between 0 and 1) and one saddle point, which is enough to determine that there are at least 3 critical points, which means the derivative of the function is at least of degree 3, which would in turn mean the original function would have degree at least 4.
