Final answer:
None of the provided options correctly represent a second-degree polynomial function with a leading coefficient of -1 and a root of 4 with multiplicity 2. The correct function should take the form f(x) = -x² + 8x - 16, which is not listed among the options.
Step-by-step explanation:
The student is looking for a second-degree polynomial function with a leading coefficient of -1 and a root of 4 with multiplicity 2. A polynomial function with a root of 4 with multiplicity 2 means that (x - 4) will be a factor of the function, squared because of the multiplicity of 2. This root squared would be represented as (x - 4)².
Considering the leading coefficient of -1, the polynomial would have the form f(x) = -1×(x - 4)². Expanding this, we obtain f(x) = -1 × (x² - 8x + 16), which simplifies to f(x) = -x² + 8x - 16. Among the given options, Option C f(x) = -x² - 8x + 16 is incorrect because of the sign in front of 8x, therefore the correct answer must be an option not listed here as none of the options accurately reflects a polynomial function with these characteristics.