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Let's break down the given expression and solve it step by step.

The expression to multiply is:

\[
\frac{x^2 - 2x - 3}{x + 2} \cdot \frac{3x + 6}{x^2 - 1}
\]

### Step 1: Factor the expressions
We will factor each part where possible:

1. **Factor \( x^2 - 2x - 3 \)**:
   \[
   x^2 - 2x - 3 = (x - 3)(x + 1)
   \]
2. **Factor \( 3x + 6 \)**:
   \[
   3x + 6 = 3(x + 2)
   \]
3. **Factor \( x^2 - 1 \)** (difference of squares):
   \[
   x^2 - 1 = (x - 1)(x + 1)
   \]

### Step 2: Substitute the factored expressions into the original equation:
Now substitute the factored expressions into the original fraction:

\[
\frac{(x - 3)(x + 1)}{x + 2} \cdot \frac{3(x + 2)}{(x - 1)(x + 1)}
\]

### Step 3: Cancel out common terms
- The factor \( (x + 2) \) appears in both the numerator and denominator, so we can cancel it out.
- The factor \( (x + 1) \) also appears in both the numerator and denominator, so we cancel it as well.

After canceling, we are left with:

\[
\frac{x - 3}{x - 1} \cdot 3 = \frac{3(x - 3)}{x - 1}
\]

### Final Answer:
The simplified expression is:

\[
\frac{3(x - 3)}{x - 1}
\]

Would you like further details on any part of this process?

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Here are 5 related questions for further practice:

1. How do you factor quadratic expressions like \(x^2 - 2x - 3\)?
2. What is the difference of squares method in factoring?
3. Can you simplify expressions with multiple variables in a similar way?
4. How do you multiply and divide rational expressions in algebra?
5. What are common mistakes to avoid when simplifying rational expressions?

**Tip**: Always check for common factors in both the numerator and denominator before simplifying!

Multiply the expressions: \(\frac{x^2 - 2x - 3}{x + 2} \cdot \frac{3x + 6}{x^2 - 1}\), and simplify the answer as much as possible.

This page covers the multiplication and simplification of rational expressions involving quadratic factoring. Learn how to factor quadratic expressions and apply the difference of squares method to simplify the expression \(\frac{x^2 - 2x - 3}{x + 2} \cdot \frac{3x + 6}{x^2 - 1}\). Suitable for students in Grades 9-11.

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