Step 1:

Assume that he equation is has more than two real roots.

Notice that polynomials are differentiable and  continuous over the reals.

By Rolls theorem :

If a function, , is continuous on the closed interval , is differentiable on the open interval , and , then there exists at least one number , in the interval such that .

Consider .

The function has exactly one critical point, at .

Therefore, it is a decreasing function for and an increasing function for .

If , then for all x, and hence it has no real roots.

If , then has a single real zero at .

If , then Find x-values to the left and right of x = -1 where f > 0 and use the Intermediate Value Theorem to infer that f has two real roots. We\\'ve covered all possible cases for c, and we find that we never have more than two real roots.