Answer
The equation of the parabola is $$ x^{2}+4x+8y-4=0 $$
Work Step by Step
Given
Vertex $
(-2,1)$, Focus $(-2,-1)$
From the coordinates of the vertex and focus, we find the axis of the parabola is parallel to the $y$-axis $(x=-2)$ and opens downward.
So, the general equation of the parabola is
$$ (x-h)^{2}=4 p(y-k) $$
So $(h,k)=(-2,1)$ and $p=-1-1=-2$
And
$$ (x+2)^{2}=4 (-2)(y-1) $$
so we get $$ x^{2}+4x+8y-4=0 $$
