23: The derivative
has exactly one root. Therefore f itself can have at most two roots. The function goes to infinity at both infinity and minus infinity, so it may have no, one or two roots. One way to decide is to look at the position of its local (and hence global) minimum. The above derivative yields one critical point, substituting it into f it is easy to see that the value at this minimum is less than zero. This implies exactly two roots.
Now use the Intermediate value theorem to identify their positions, try substituting nice integer values into f and wait for a sign change between two successive ones.