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.