Answer
The solution set is $\left\{-5-\sqrt{10}, -5+\sqrt{10}\right\}$.
Work Step by Step
Subtract $21$ from each side to obtain:
$$-3(y+5)^2=-30$$
Divide $-3$ to both sides to obtain:
$$(y+5)^2=10$$
Take the square root of both sides to obtain:
\begin{align*}
\sqrt{(y+5)^2}&=\pm\sqrt{10}\\
y+5&=\pm\sqrt{10}\\
y&=-5\pm\sqrt{10}
\end{align*}
Checking:
\begin{align*}
-3(-5-\sqrt{10}+5)^2+21&=-9\\
-3(-\sqrt{10}^2+21&=-9\\
-3(10)+21&=-9\\
-30+21&=-9\\
-9&\stackrel{\checkmark}=-9
\end{align*}
\begin{align*}
-3(-5+\sqrt{10}+5)^2+21&=-9\\
-3(\sqrt{10}^2+21&=-9\\
-3(10)+21&=-9\\
-30+21&=-9\\
-9&\stackrel{\checkmark}=-9
\end{align*}
Therefiore, the solution set is $\left\{-5-\sqrt{10}, -5+\sqrt{10}\right\}$.
