because we have the points (-2,1) and (4,-5), these are the x-values and corresponding y-values for two points on our parabola.
so we have
now, since the line x = 0 is the axis of symmetry, it means the vertex is on this line. so the x-coordinate for the vertex is 0. since the vertex is (h,k), this means
thus you have 3 equations, 3 unknowns, you can solve for the other 2 unknowns to get your quadratic.
it may do you good to see the Conic sections link in my signature
You've studied vertices and axes of symmetry in your class, so you know that, if the axis of symmetry is the line x = 0 (that is, the y-axis), then the vertex is of the form (h, k) = (0, k) for some value of "k".
Have you not studied systems of linear equations, so that's where you're bogging down?