To find \( F(x + h) \) for the function \( F(x) = x^2 - 6x \), we need to substitute \( x + h \) into the function for \( x \).

### Steps:

1. Start with the original function:
   \[
   F(x) = x^2 - 6x
   \]

2. Substitute \( x + h \) in place of \( x \):
   \[
   F(x + h) = (x + h)^2 - 6(x + h)
   \]

3. Now expand the terms:

   - \( (x + h)^2 = x^2 + 2xh + h^2 \)
   - \( -6(x + h) = -6x - 6h \)

   So, we get:
   \[
   F(x + h) = x^2 + 2xh + h^2 - 6x - 6h
   \]

### Final answer:
\[
F(x + h) = x^2 + 2xh + h^2 - 6x - 6h
\]

Would you like further details or clarifications?

Here are some related questions to explore:

1. How do you simplify expressions involving \( F(x + h) - F(x) \)?
2. What is the significance of \( h \) approaching 0 in limits?
3. Can you explain the process for finding the derivative using the definition of a derivative?
4. How does the chain rule apply to composite functions like \( F(g(x)) \)?
5. What are the steps to expand binomials like \( (x + h)^2 \)?

**Tip:** When working with functions and limits, expanding and simplifying expressions carefully is key to avoiding mistakes.

Learn how to expand the function F(x) = x^2 - 6x by finding F(x + h). This example demonstrates substitution, binomial expansion, and simplifying algebraic expressions. Suitable for high school students in Grades 9-12.

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