an ellipse is the locus of all points whose sum of distances to the foci is a constant

the verticies are the top and bottom of the ellipse on the vertical line x=3. Center of the ellipse is (3,2) midway between the two verticies. The foci are on a horizontal line y=2

any point on the ellipse, (x,y) is equal to the sum of distances to the foci, (-1,2) and (7,2)

that distance is 10. To get 10 from horizontal vertex to sum of distances to both foci, the horizontal co-verticies must be (-2,2) and (8,2)

(x-h)^{2}/a^{2} + (y-k)^{2}/b^{2} = 1, a=5, b=3 (h,k) is the center (3,2) a is the distance from the center to the left vertex and from the center to the right vertex b is the distance from the center to the top vertex and from the center to the bottom vertex

(x-3)^{2}/25 + (y-2)^{2}/9 = 1

or multiplying through by 25 times 9

9(x-3)^{2} + 25(y-2)^{2} = 225

If you want you can expand that further and combine constant terms to get

9x^{2}-54x +25y^{2} - 100y -44 = 0

but the first equation gives you the most information immediately about the vertices

The graph is a flattened ellipse, longer horizontally than vertically