The question asks to simplify the following expression:

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

To solve this, let's go through the steps:

Step 1: Simplify Each Fraction

1. First Fraction: $\frac{x^2 + x - 2}{x^2 - 3x + 2}$

   • Factor the numerator: $x^2 + x - 2 = (x + 2)(x - 1)$
   • Factor the denominator: $x^2 - 3x + 2 = (x - 1)(x - 2)$

   So, we can rewrite this fraction as: $\frac{(x + 2)(x - 1)}{(x - 1)(x - 2)}$

   Cancel out $(x - 1)$ from the numerator and denominator: $\frac{x + 2}{x - 2}$

2. Second Fraction: $\frac{2x + 4}{x^2 - 4}$

   • Factor the numerator: $2x + 4 = 2(x + 2)$
   • Factor the denominator (difference of squares): $x^2 - 4 = (x + 2)(x - 2)$

   So, this fraction becomes: $\frac{2(x + 2)}{(x + 2)(x - 2)}$

   Cancel out $(x + 2)$ from the numerator and denominator: $\frac{2}{x - 2}$

Step 2: Divide the Fractions

Now we have: $\frac{x + 2}{x - 2} \div \frac{2}{x - 2}$

Dividing by a fraction is equivalent to multiplying by its reciprocal: $\frac{x + 2}{x - 2} \times \frac{x - 2}{2}$

Cancel $(x - 2)$ from the numerator and denominator: $\frac{x + 2}{2}$

Final Answer

The simplified form of the expression is: $\frac{x + 2}{2}$

Would you like a detailed explanation of each factoring step, or do you have any specific questions?

Related Questions

1. How do you simplify expressions involving division of rational functions?
2. What are the methods for factoring quadratic expressions?
3. How do you handle division by a fraction in algebra?
4. What is the process for simplifying complex rational expressions?
5. How does factoring help in simplifying algebraic fractions?

Tip

When simplifying rational expressions, always look for common factors in the numerator and denominator to cancel out terms easily.