Write the equation of a hyperbola with vertices (3, -1) and (3, -9) and co-vertices (-6. -5) and (12, -5).

The
the main formula of hyperbola is

(x-h)² / a² - (y-k)² / b² = 1, where the center is C (h,k)
and the vertex formmula is

(h+-a, k) and the covertex is (h, k+-b)

in our case vertices are already given:
(3, -1) and (3, -9)  so h+a =3  or  h - a =3 and k= -1, or k= -9
and co-vertices (-6. -5) and (12, -5)
h= -6  or h= 12, and  k+ b=-5, or k- b = -5
for example h= -6, h+a =3, a=3+6=9, k= -1, k+ b=-5 , we can get b= - 5 + 1 = - 4, so a=9, b=-4, h=-6, k= -1
the equation is
(x+6)² / 81 -  (y+1)² / 16 = 1, if the center is C(-6, -1)

the same method can be applied with the other choices of h and k.