(-3,0) and (5,0) are on your parabola i.e. if you plug in -3 and -5 into your equation, you'll get 0.
Now looking at your equation:
The only way that your equation will yield 0 is if either the red or blue is equal to 0 (remember, : either a or b or both are 0). Since we know x = -3 and 5 will work, we can see that r = -3 and s = 5 (since 3 - (-3) = 0 and 5 - 5 = 0).
Now all that's left to do is solve for a. You have the point (1, -4) - i.e. if x = 1 and you plug it into your equation, you'll get y = -4. So plug this into your expression from above and you'll get your parabola.