What is the equation of the quadratic function with roots -3 and 1 and a vertex a (-1, -8)?

Start with the equation of a vertical parabola:  y-k = a(x-h)^2.  We are told that the coordinates of the vertex are -1 and -8.  Substituting these into the above equation,

y-[-8] = a(x-[-1]), or y+8 = a(x+1)^2.  The coefficient a is undetermined here.  How do we find it?

At any root, y=0, as a "root" is a point at which the graph crosses the x-axis.

Thus, if we choose to work with the given root 1, the associated point is (1,0).  Substitute these coordinates into the above equation y+8 = a(x+1)^2.

We get 0+8 = a(1+1)^2, or 8 = a(2)^2, or 8 = 4a, or a = 2.

Substituting this value for a into the above equation,

y+8 = 2(x+1)^2.

You should check this result by determining whether the other root (-3,0) satisfies this equation.  If yes, then you'll know for certain that this equation correctly describes the parabola in question.