jmc

The radical conjugate of $a+\sqrt{b}$ is $a-\sqrt{b}$. Hence their sum is $2a$. Then we know that $2a=-4$ which gives us $a=-2$. The product $(a+\sqrt{b})\cdot (a-\sqrt{b})=a^2-b=1.$ Plugging in the value for $a$, we can solve for $b$ to get that $b=(-2{)}^2-1=3$. Therefore $a+b=-2+3=\boxed{1}$.