Question

find the equation for an ellipse that satisfies the following condition
Vertices at (-3,1) and (9,1)
one focus at (7,1)

I will assume you know the general equation of an ellipse and its properties in terms of a,b and c.

the centre would be the midpoint of (-3,1) and (9,1) which is (3,1) from which we can easily see that a = 6
one focal point is (7,1) so the distance from the centre to the focal point is 4
therefore c=4

now in a horizontally placed ellipse
b^2 + c^2 = a^2
.
.
.
.

Your equation should be

(x-3)^2/ 36 + (y-1)^2 /48 = 1