Answer
$${\left( {x + 2} \right)^2} = - 16\left( {y - 1} \right)$$
Work Step by Step
$$\eqalign{
& {\text{vertex }}\left( { - 2,1} \right),\,\,{\text{focus }}\left( { - 2, - 3} \right) \cr
& {\text{Because the focus is downward the vertex}} \cr
& {\text{The equation is of the form }}{\left( {x - h} \right)^2} = 4p\left( {y - k} \right) \cr
& {\text{With vertex }}\left( {h,k} \right){\text{ and focus }}\left( {h,p + k} \right) \cr
& h = - 2,\,\,k = 1,\,\,\,\,p + 1 = - 3,\,\,\,p = - 4 \cr
& \cr
& {\text{The equation is }} \cr
& {\left( {x - h} \right)^2} = 4p\left( {y - k} \right) \cr
& {\left( {x + 2} \right)^2} = 4\left( { - 4} \right)\left( {y - 1} \right) \cr
& {\left( {x + 2} \right)^2} = - 16\left( {y - 1} \right) \cr} $$
