Construct a polynomial function with the following properties: fifth degree, 4 is a zero of multiplicity $4,-2$ is the only other zero, leading coeficient is 5 . Answer \[ f(x)= \]

Step 1 ：We are asked to construct a polynomial function with the following properties: it is of fifth degree, 4 is a zero of multiplicity 4, -2 is the only other zero, and the leading coefficient is 5.

Step 2 ：A polynomial function of degree n can have at most n zeros. Since the polynomial function we are constructing is of degree 5, it can have at most 5 zeros.

Step 3 ：We are given that 4 is a zero of multiplicity 4 and -2 is the only other zero. This means that the polynomial function can be written in the form of \(f(x) = a(x - 4)^4(x + 2)\), where a is the leading coefficient.

Step 4 ：We are given that the leading coefficient is 5, so the polynomial function is \(f(x) = 5(x - 4)^4(x + 2)\).

Step 5 ：Expanding this expression, we get \(f(x) = 5x^5 - 70x^4 + 320x^3 - 320x^2 - 1280x + 2560\).

Step 6 ：\(\boxed{f(x) = 5x^5 - 70x^4 + 320x^3 - 320x^2 - 1280x + 2560}\) is the polynomial function with the given properties.