Function $f$ is defined by $f(x)=(x-3)(x+1)$. Function $g$ is defined by $g(x)=f(x+2)$. The graph of $y=g(x)$ in the $xy$-plane has $x$-intercepts at $(a,0)$ and $(b,0)$, where $a$ and $b$ are distinct constants. What is the value of $a+b$?