To determine which second-degree polynomial function fits the given conditions, we need to follow these steps carefully:
### 1. Understanding the Conditions
We are given:
- The leading coefficient of the polynomial is -1.
- The polynomial has a root of 4 with multiplicity 2.
### 2. Formulating the Polynomial
A root of 4 with multiplicity 2 implies that the polynomial can be written in the form:
[tex]\[ f(x) = -1 \cdot (x - 4)^2 \][/tex]
### 3. Expanding the Polynomial
Next, we need to expand the polynomial [tex]\((x - 4)^2\)[/tex].
[tex]\[
(x - 4)^2 = (x - 4)(x - 4) = x^2 - 8x + 16
\][/tex]
### 4. Applying the Leading Coefficient
Now we multiply by the leading coefficient -1:
[tex]\[
f(x) = -1 \cdot (x^2 - 8x + 16)
\][/tex]
Distribute the -1 across the terms in the polynomial:
[tex]\[
f(x) = -x^2 + 8x - 16
\][/tex]
### 5. Comparing with the Given Options
Now we need to compare our derived polynomial [tex]\(-x^2 + 8x - 16\)[/tex] with the given options:
1. [tex]\( f(x) = -x^2 - 8x - 16 \)[/tex]
2. [tex]\( f(x) = -x^2 + 8x - 16 \)[/tex]
3. [tex]\( f(x) = -x^2 - 8x + 16 \)[/tex]
4. [tex]\( f(x) = -x^2 + 8x + 16 \)[/tex]
### 6. Choosing the Correct Option
The correct polynomial from our derivation is:
[tex]\[ -x^2 + 8x - 16 \][/tex]
Comparing this with the provided options, it matches the second option:
[tex]\[
f(x) = -x^2 + 8x - 16
\][/tex]
### Conclusion
Therefore, the correct second-degree polynomial function is:
[tex]\[
\boxed{-x^2 + 8x - 16}
\][/tex]
