Let's work through both parts of the problem, step by step.

Given:

$f(x) = 7 - x^2$

(a) $f(x + h) - f(x)$

1. Find $f(x + h)$:
   Replace $x$ with $x + h$ in the function $f(x)$. $f(x + h) = 7 - (x + h)^2$ Expanding the square: $f(x + h) = 7 - (x^2 + 2xh + h^2) = 7 - x^2 - 2xh - h^2$

2. Subtract $f(x)$ from $f(x + h)$: $f(x + h) - f(x) = \left( 7 - x^2 - 2xh - h^2 \right) - (7 - x^2)$ Simplifying: $f(x + h) - f(x) = 7 - x^2 - 2xh - h^2 - 7 + x^2$ Cancel out terms: $f(x + h) - f(x) = -2xh - h^2$

Thus, the simplified expression for $f(x + h) - f(x)$ is: $f(x + h) - f(x) = -2xh - h^2$

(b) $\frac{f(x + h) - f(x)}{h}$

Now, take the result from part (a) and divide by $h$:

$\frac{f(x + h) - f(x)}{h} = \frac{-2xh - h^2}{h}$

Simplify by dividing each term by $h$: $\frac{f(x + h) - f(x)}{h} = -2x - h$

Thus, the simplified expression for $\frac{f(x + h) - f(x)}{h}$ is: $\frac{f(x + h) - f(x)}{h} = -2x - h$

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Would you like further details or have any questions? Here are some follow-up questions for practice:

1. What would $f(x + h) - f(x)$ look like for a different quadratic function?
2. How would the expression change if the original function were linear?
3. What happens to the result in part (b) as $h \to 0$?
4. Can you find the derivative of $f(x) = 7 - x^2$ using part (b)?
5. How does the process change if $f(x)$ is cubic rather than quadratic?

Tip: When simplifying expressions, always look for common terms to cancel out before dividing.