Let's work through the problem step by step.

We are given: $f(x) = 8 - x^2$ and need to simplify: $f(x+h) - f(x)$

Step 1: Find $f(x+h)$

Substitute $x+h$ into the function $f(x)$: $f(x+h) = 8 - (x+h)^2$

Step 2: Expand $(x+h)^2$

$(x+h)^2 = x^2 + 2xh + h^2$ Thus, $f(x+h) = 8 - (x^2 + 2xh + h^2) = 8 - x^2 - 2xh - h^2$

Step 3: Subtract $f(x)$ from $f(x+h)$

Now subtract $f(x) = 8 - x^2$ from $f(x+h)$: $f(x+h) - f(x) = \left(8 - x^2 - 2xh - h^2\right) - (8 - x^2)$

Simplify the expression: $f(x+h) - f(x) = 8 - x^2 - 2xh - h^2 - 8 + x^2$

The $8$ and $-8$ cancel, and the $-x^2$ and $x^2$ cancel, leaving: $f(x+h) - f(x) = -2xh - h^2$

Final Answer:

$f(x+h) - f(x) = -2xh - h^2$

Would you like more details on any part of the solution?

Here are 5 related questions:

1. How would this expression be used to find the difference quotient?
2. How does the expansion of $(x+h)^2$ affect the final expression?
3. What is the geometric interpretation of $f(x+h) - f(x)$?
4. How do we generalize this process for other types of functions?
5. How would this approach change for a cubic function?

Tip: When simplifying expressions like $f(x+h) - f(x)$, be careful with distributing negative signs and combining like terms to avoid errors.