Just take the first and second derivatives and solve them for f(x) = 0.
You will note that you will have either:
a) different values for the maximum, minimum and inflection point
b) identical values for the max, min and inflection point
For case A, you can interpret it as the function having two "changes of direction" (at the maximum point the function stops increasing and starts decreasing, and inversely for the minimum point). Even if both maximum and minimum points have the same sign, there has to be at least one intersection with the x-axis, either before or after the interval between the maximum and minimum points. If they have opposite signs, then there may be a root before, after and between the maximum and minimum interval.
For case B, all three points are the same so after that point the function will continue its original course so there must be one intersection anyway.
Alternatively (and maybe more rigurously) as EVERY -linear- cubic function (as the one stated in your question) can be represented as the product of a first and second order polynomial, non real roots can only be obtained from the second order polynomial. The root of the first order polynomial will always be real.