Surely you know that any quadratic function can be written as y= a(x- b)(x- c) where b and c are the zeros of the function.. Knowing that x= -1 is a zero tells us that we can write y= a(x- b)(x+ 1)
Knowing that (-3, 4) is the vertex we know two things.
First we know that y(-3)= a(-3- b)(-3+ 1)= 2a(b+ 3)= 4.
Second, we know that the slope of the graph at x= -3 is 0. How you use that depends upon what you have learned. If you have studied Calculus, the derivative of y, y'= a(x+ 1)+ a(x- b) is 0 when x= -3. That is, y'(-3)= -2a+ a(-3- b)= 0 so -5a- ab= -a(b+ 5)= 0. Either a= 0 or b= -5. If a= 0, 2a(b+ 3)= 4 cannot be true so we must have b= -5. The other root is x= -5.
If you don't know Calculus, you can use the fact that vertex of a parabola is always exactly half way between its zeros! Here, one root is at x= -1 and the vertex is at x= -3. -1= -3+ 2 so the other root is at x= -3-2= -5.
Lacking either of those methods, you can "complete the square". If we write y= p(x- q)^2+ b then when x= q, y= b while for any other x, y is b plus or minus something, depending on the sign of p. That is, (q, b) is the vertex. Since we are told that (-3, 4) is the vertex, y= p(x+ 3)^2+ 4. Setting x= -1 y= 4p+ 4= 0 so p= -1. The formula is y= -(x+ 3)^2+ 4= 0 so (x+ 3)^2= 4, x+3= +/- 2. x= -3+ 2= -1 and x= -3- 2= -5.
