Which polynomial function has a leading coefficient of 1, roots –3 and 8 with multiplicity 1, and root 4 with multiplicity 2?

It will  be a  polynomial  of degree 4  because it has 4 roots.
written as factors it is

(x + 3)(x - 8)(x - 4)^2

expanding:-

= (x^2 -5x - 24)(x^2 - 8x + 16)

= x^4 - 8x^3 + 16x^2 - 5x^3 + 40x^2  - 80x - 24x^2 + 192x - 384

= x^4 - 13x^3 + 32x^2 + 112x - 384