Answer
$f(x) =-2\left(x+3\right)^2-1$
Work Step by Step
Given
\begin{equation}
f(x)=a(x-h)^2+k
\end{equation}
We know that the vertex is $(h,k) = (-3,-1)$. This means that
$f(x) =a(x+3)^2-1 $.
Since the point $(-2,-3)$ lies on the graph, it follows that:
\begin{equation}
\begin{aligned}
a(-2+3)^2-1 & = -3\\
a&= -3+1\\
a& = -2 \\
\end{aligned}
\end{equation} Hence
$$
f(x) =-2\left(x+3\right)^2-1
$$
