jmc

Since both of these points lie on the line, plugging them into the equation of the line will produce a true statement. Thus $(-2,0)$ gives us $0=-2m+b$ and $(0,2)$ gives us $2=b$. So we now know what $b$ is and can plug it back into the first equation to get $0=-2m+2$. So $m=1$ and $m+b=\boxed{3}$.