You are given that a quadratic function is a parabola with vertex (-4,0).


It means that the function in vertex form is  

    y = a*(x-(-4))^2 + 0 = a*(x+4)^2.      (1)


The only unknown is the parameter "a". To find it, use the given part, which says
that the parabola passes through the point (1,-75).


So, at x= 1 the value of the function (1) should be -75.  You write this equation, based on (1)

    -75 = a*(1+4))^2,  or  -75 = a*5^2,  which is  25a = -75.


From this equation,  a =  = -3.   


ANSWER.  The value of "a" is  -3.

That's ABSOLUTELY unnecessary to complete the square and go through all that "mumbo-jumbo" as the other person did.
This is done as easily as: Vertex form of the equation 
                            of a parabola: 
                                         ------ Substituting (1, - 75) for (x, y) and (- 4, 0) for (h, k)
                                         - 75 = 25a
                                       
                                       That's ALL!!