If your zeros are 2, -2, and 1, your polynomial can be written as y=(x-2)(x+2)(x-1)-a

Since the y-intercept is 24, then 24=(0-2)(0+2)(0-1)-a

So 24=4-a, and so a=-20

Your polynomial would be y=(x-2)(x+2)(x-1)-20, and just expand the (x-2)(x+2)(x-1) part to get your final polynomial.