Answer
$$\frac{{{x^2}}}{{12}} + \frac{{{y^2}}}{{16}} = 1$$
Work Step by Step
$$\eqalign{
& {\text{ellipse; vertex at }}\left( {0, - 4} \right),{\text{ focus at }}\left( {0, - 2} \right) \cr
& {\text{The }}x{\text{ - coordinate are the same, then the ellipse}} \cr
& {\text{has the equation }}\frac{{{x^2}}}{{{b^2}}} + \frac{{{y^2}}}{{{a^2}}} = 1 \cr
& {\text{With:}} \cr
& {\text{vertices }}\left( {0, \pm a} \right) \to \,\,\,a = 4 \cr
& {\text{foci: }}\left( {0, \pm c} \right) \to c = 2 \cr
& {b^2} = {a^2} - {c^2} \cr
& {b^2} = {4^2} - {2^2} \cr
& {b^2} = 12 \cr
& \cr
& {\text{The equation of the ellipse is}} \cr
& \frac{{{x^2}}}{{12}} + \frac{{{y^2}}}{{16}} = 1 \cr} $$
