Well it depends on how you count.
The Fundamental Theorem of Algebra is that every nth degree polynomial equation has exactly n complex roots, counting multiplicities. So for our cubic the answer is
Answer: 3
We have to check to see if the roots are duplicated. We can see this one has three unique real roots. The easiest way is to look at
f(-1) = -4
f(0) = 1
f(1) = -2
f(3) = 16
We see there's a real zero between x=-1 and 0, between 0 and 1 and between 1 and 3.
The question might be asking how many complex roots which aren't real are there, in which case the answer is zero.