Take two numbers, say -1 and 0. Then f(-1) = -6 < 0 and f(0) = 1 > 0 (Letting f(x) = 1 + 2x + x^3 + 4x^5.)
By the Intermediate Value Theorem there exists a number "c" between -1 and 0 such that f(c) = 0. So the equation has a real root. Then suppose there are two roots a and b.
Then f(a)...