Counting real roots

Derivative. $f'(x)=3x^2-3=3(x-1)(x+1)$.

Critical points. $f'$ vanishes at $x=-1$ and $x=1$. $f'<0$ on $]-1,1[$ (so $f$ decreases), $f'>0$ outside.

Values. $f(-1)=-1+3+1=3>0$; $f(1)=1-3+1=-1<0$.

Limits. $\lim_{-\infty}f=-\infty$, $\lim_{+\infty}f=+\infty$.

Conclusion. By the intermediate value theorem, $f$ has one root in $]-\infty,-1[$, one in $]-1,1[$, one in $]1,+\infty[$. Three distinct real roots.