What is a quartic polynomial function in standard form with zeros - 2,4,4, and 3

We can simply multiply the roots together to find the original function.

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

FOIL.

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

Combine like terms.

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

FOIL.

x^3 - 2x^2 - 8x - 4x^2 + 8x + 32 (x - 3)

Combine like terms.

x^3 - 6x^2 + 32 (x - 3)

FOIL.

x^4 - 6x^3 + 32x - 3x^3 + 18x^2 - 96

Combine like terms.

x^4 - 9x^3 + 18x^2 + 32x - 96 is the original function with the given roots.