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We are given two functions:

\[
f(x) = -5x^2 + 8x - 7
\]
\[
g(x) = x^2 - 1
\]

The question asks for the expression equivalent to \( f(x) - g(x) \). Let's find this step by step.

### Step 1: Write the expression for \( f(x) - g(x) \)

\[
f(x) - g(x) = (-5x^2 + 8x - 7) - (x^2 - 1)
\]

### Step 2: Distribute the negative sign across \( g(x) \)

\[
f(x) - g(x) = -5x^2 + 8x - 7 - x^2 + 1
\]

### Step 3: Combine like terms

\[
(-5x^2 - x^2) + 8x + (-7 + 1)
\]

\[
= -6x^2 + 8x - 6
\]

Thus, the expression for \( f(x) - g(x) \) is:

\[
-6x^2 + 8x - 6
\]

### Final Answer:
The correct option is **C) \(-6x^2 + 8x - 6\)**.

Let me know if you'd like further clarification! 

### Additional Questions:
1. How do you subtract two functions algebraically in general?
2. Can you explain how the distributive property works in subtraction?
3. What are "like terms" in polynomials and why do we combine them?
4. How can you check your result after finding the difference between two functions?
5. What are some other types of operations you can perform on functions (addition, multiplication, division)?

**Tip:** When subtracting two functions, always distribute the negative sign carefully across all terms to avoid mistakes.

Given f(x) = -5x^2 + 8x - 7 and g(x) = x^2 - 1, which of the following expressions is equivalent to f(x) - g(x)?

Learn how to subtract two polynomial functions f(x) = -5x^2 + 8x - 7 and g(x) = x^2 - 1. Follow a detailed step-by-step explanation, suitable for Grades 9-10, covering algebraic concepts and function operations.

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