y = a(x – h)^2 + k      <<<---=== it is the (general) vertex form of a quadratic function

y = a(x +7)^2- 9        <<<---=== it is the vertex form of a GIVEN quadratic function written specifically for the GIVEN vertex.
                                  The coefficient "a" still remains undetermined.
                                  And our goal is to determine it.

0 = a(-10 +7)^2 - 9     <<<---=== To determine the coefficient "a", use the condition that x= -10 is the root.
                                  At this point my solution becomes different from @richwmiller writing, who made a mistake in this point.

0 = a(-3)^2 - 9

9 = a(-3)^2

9 = 9a  ====>

a =  = 1.


Thus (and finally), your function is

y = (x + 7)^2 - 9


Check.  a)  (-7,-9) is the vertex; it is CLEAR.

         b) the function is ZERO at x= -10:  y = (-10+7)^2 - 9 = 9 - 9 = 0.

                             and at x = -4:  y = (-4-7)^2 - 9  = 9 - 9 = 0.

Answer.  Your function is  y = (x+7)^2 - 9.