f(x) = x^{4}-4x^{3}-8x^{2}+4

How many times does f(x) = 0? (excuse the bad formulation)

Hint: Gather as much information from the derivative of f(x) as possible. We are only interested in the number of times f(x) equals zero.

Okay so differentiating the function is simple: f ' (x) = 4x^{3}-12x^{2}-16x, but what kind of information do I need to extract in order to determine how many times f(x)=0?