### Instruction: The number $a+\sqrt{b}$ and its radical conjugate have a sum of $-4$ and a product of $1$. Find $a+b$.### Response: I know that the radical conjugate of $a+\sqrt{b}$ is $a-\sqrt{b}$, so I can write the equations for their sum and product as follows:

\begin{align*}
(a+\sqrt{b}) + (a-\sqrt{b}) &= -4 \\
(a+\sqrt{b}) (a-\sqrt{b}) &= 1
\end{align*}
Simplifying the first equation, I get $2a = -4$, so $a = -2$.