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We are tasked with finding \((g - f)(-2)\), where:

\[
f(x) = x^2 - 10 \quad \text{and} \quad g(x) = 5x + 13
\]

The operation \((g - f)(x)\) is defined as the difference between the functions \(g(x)\) and \(f(x)\), which is:

\[
(g - f)(x) = g(x) - f(x)
\]

### Step 1: Set up \((g - f)(x)\)
\[
(g - f)(x) = (5x + 13) - (x^2 - 10)
\]
Simplifying:
\[
(g - f)(x) = 5x + 13 - x^2 + 10
\]
\[
(g - f)(x) = -x^2 + 5x + 23
\]

### Step 2: Evaluate at \(x = -2\)
Now, substitute \(x = -2\) into \((g - f)(x)\):
\[
(g - f)(-2) = -(-2)^2 + 5(-2) + 23
\]
\[
(g - f)(-2) = -(4) - 10 + 23
\]
\[
(g - f)(-2) = -4 - 10 + 23
\]
\[
(g - f)(-2) = 9
\]

Thus, the answer is:
\[
(g - f)(-2) = 9
\]

Therefore, the correct choice is **A**.

---

Do you want further details or have any questions?

Here are 5 related questions:
1. How do we generally perform function subtraction like \((g - f)(x)\)?
2. What happens if \(x^2 - 10\) had been \(x^2 + 10\)?
3. How would the result change if \(g(x) = 3x + 15\) instead of \(5x + 13\)?
4. What is the domain of the function \((g - f)(x)\)?
5. How can you check if a result like \((g - f)(-2)\) exists for a given function?

**Tip:** When performing function subtraction, remember to distribute any negative signs carefully to avoid mistakes.

Given f(x) = x^2 - 10 and g(x) = 5x + 13, find (g - f)(-2), if it exists.

Learn how to find (g - f)(-2) given the functions f(x) = x^2 - 10 and g(x) = 5x + 13. This problem involves function subtraction, quadratic, and linear functions. Suitable for high school students in Grades 9-10.

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