Find an equation for the conic section with the given properties.

(-3,-2)    (7,-2)       this will  be  a parabola that opens sideways

2a  =the distance between  the  vertices

2a = (7 - -3)  = 10

a  = 5   .....   a^2  = 25

2c = the distance between the foci  

2c = (12 - -8)

2c  = 20

c =10   .....     c^2  =100    

c^2 - a^2  =  b^2 

b^2  =  10^2 - 5^2  =  75

The center is  [ (-3 + 7)/2 , -2 ]  =  [ 4/2, -2]=  (2, -2)  =(h , k)

The equation  is

( x -h)^2         (y - k)^2

_______  -    ________  =      1

    a^2                b^2

( x - 2)^2        ( x + 2)^2

_______  -   _________  =    1

    25              75

Here's a graph :  https://www.desmos.com/calculator/jkmg428rig