Answer
$${x^2} - 4x - 8y - 44 = 0$$
Work Step by Step
$$\eqalign{
& {\text{Vertex: }}\left( {2,6} \right) \cr
& {\text{Focus }}\left( {2,4} \right) \cr
& {\text{The }}x{\text{ - coordinate is the same, so the equation of the parabola}} \cr
& {\text{is of the form }}{\left( {x - h} \right)^2} = 4p\left( {y - k} \right) \cr
& {\text{With:}} \cr
& {\text{Vertex }}\left( {h,k} \right):\left( {2,6} \right) \to h = 2,{\text{ }}k = 6 \cr
& {\text{Focus }}\left( {h,p + k} \right):\left( {2,4} \right) \to p + k = 4,{\text{ }}p + 6 = 4,{\text{ }}p = - 2 \cr
& \cr
& \underbrace {{{\left( {x - h} \right)}^2} = 4p\left( {y - k} \right)}_ \downarrow \cr
& {\left( {x - 2} \right)^2} = 4\left( { - 2} \right)\left( {y - 6} \right) \cr
& {\left( {x - 2} \right)^2} = - 8\left( {y - 6} \right) \cr
& {\text{Expand}} \cr
& {x^2} - 4x + 4 = - 8y + 48 \cr
& {x^2} - 4x - 8y - 44 = 0 \cr} $$
