Answer
The rectangular equation is, ${{x}^{2}}+{{\left( y+2 \right)}^{2}}=4$
Work Step by Step
Using, ${{r}^{2}}={{x}^{2}}+{{y}^{2}}$, we will convert the polar equation to the rectangular equation.
Therefore,
$\begin{align}
& r=-4\sin \theta \\
& {{r}^{2}}=-4r\sin \theta \\
& {{x}^{2}}+{{y}^{2}}=-4r\sin \theta \\
& =-4y
\end{align}$
We can further simplify it by completing the square on y as,
$\begin{align}
& {{x}^{2}}+{{y}^{2}}=-4y \\
& {{x}^{2}}+{{y}^{2}}+4y+4=4 \\
& {{x}^{2}}+{{\left( y+2 \right)}^{2}}=4
\end{align}$
