jmc

The denominator, $2+4x^2$, takes all values greater than or equal to $2$. Therefore, $\frac{2}{2+4x^2}$ is at most $\frac{2}{2}=1$, and can take any positive value smaller than this. So, the range of $g(x)$ is $(0,1]$, which gives $a+b=\boxed{1}$.