Let's break down and solve the two problems presented in the image.

Problem 9:

$\frac{x^2 - 4}{x - 8} \div (x - 2)$ This is a division of rational expressions.

1. Factor the numerator of the first fraction: $x^2 - 4 = (x - 2)(x + 2)$ So the expression becomes: $\frac{(x - 2)(x + 2)}{x - 8}$

2. Rewrite the division as multiplication by the reciprocal: $\frac{(x - 2)(x + 2)}{x - 8} \times \frac{1}{x - 2}$

3. Cancel out the common factor $(x - 2)$: $\frac{x + 2}{x - 8}$

So the answer to this is $\frac{x + 2}{x - 8}$, which corresponds to option b.

Problem 10:

$\frac{s^2 - 2s}{s^2 + 3s - 10} \div \frac{s - 5}{s + 5}$ This is another division of rational expressions.

1. Factor the numerator and denominator:

   • $s^2 - 2s = s(s - 2)$
   • $s^2 + 3s - 10 = (s - 2)(s + 5)$

   So, the expression becomes: $\frac{s(s - 2)}{(s - 2)(s + 5)} \div \frac{s - 5}{s + 5}$

2. Rewrite the division as multiplication by the reciprocal: $\frac{s(s - 2)}{(s - 2)(s + 5)} \times \frac{s + 5}{s - 5}$

3. Cancel out the common factors $(s - 2)$ and $(s + 5)$: $\frac{s}{s - 5}$

So the answer to this is $\frac{s}{s - 5}$, which corresponds to option c.

Let me know if you need any further details or explanations. Here are five related questions to deepen understanding:

1. How do you handle division of rational expressions involving more complex terms?
2. What is the general process for factoring quadratic expressions?
3. How do you know when it's appropriate to cancel terms in a rational expression?
4. What happens if there is no common factor to cancel when dividing rational expressions?
5. Can these types of problems always be simplified, or are there cases where they cannot?

Tip: Always remember to factor both the numerator and denominator of expressions to spot common terms that can be canceled!