The image contains three mathematical expressions that involve division of rational functions. Let's work through them step-by-step.

(1) Expression:

$\frac{x^2 - 1}{x^2 - 4} \div \frac{x + 1}{x - 2}$ Steps:

1. Rewrite division as multiplication by the reciprocal: $\frac{x^2 - 1}{x^2 - 4} \times \frac{x - 2}{x + 1}$
2. Factor the numerators and denominators where possible: $\frac{(x - 1)(x + 1)}{(x - 2)(x + 2)} \times \frac{x - 2}{x + 1}$
3. Simplify by canceling out common terms: $= \frac{x - 1}{x + 2}$

(2) Expression:

$\frac{x - 3}{x + 3} \div \frac{x + 2}{x^2 + 2x - 3}$ Steps:

1. Rewrite division as multiplication by the reciprocal: $\frac{x - 3}{x + 3} \times \frac{x^2 + 2x - 3}{x + 2}$
2. Factor the quadratic expression: $\frac{x - 3}{x + 3} \times \frac{(x - 1)(x + 3)}{x + 2}$
3. Simplify by canceling out common terms: $= \frac{x - 3}{x + 2} \times (x - 1)$ Final result: $= \frac{(x - 3)(x - 1)}{x + 2}$

(3) Expression:

$\frac{x + 1}{x^2 - 4x + 4} \div \frac{x}{x^2 - 3x + 2}$ Steps:

1. Rewrite division as multiplication by the reciprocal: $\frac{x + 1}{(x - 2)^2} \times \frac{x^2 - 3x + 2}{x}$
2. Factor the quadratic expressions: $\frac{x + 1}{(x - 2)^2} \times \frac{(x - 1)(x - 2)}{x}$
3. Simplify by canceling out common terms: $= \frac{x + 1}{x(x - 2)}$

Let me know if you would like more detailed explanations or further clarifications!

Here are five related questions to expand on this:

1. How do you identify which terms can be factored in a rational function?
2. What rules are used to simplify expressions involving division of fractions?
3. How can common factors in the numerator and denominator affect the domain of a rational function?
4. What are the steps for performing polynomial long division when factoring isn't straightforward?
5. How can multiplying by the reciprocal help in solving complex rational equations?

Tip: Always check for any values that make the denominator zero in rational functions, as these are restrictions on the domain.