https://file.aiquickdraw.com/mathbot/file/1726573041108o7wetd3j.jpg

The expression you uploaded is:

\[
\frac{3x + 6}{x^2 - 9} \div \frac{6x^2 + 12x}{4x + 12}
\]

Let's simplify it step by step:

### Step 1: Factorize all terms.
- \( 3x + 6 \) can be factored as \( 3(x + 2) \).
- \( x^2 - 9 \) is a difference of squares: \( (x - 3)(x + 3) \).
- \( 6x^2 + 12x \) can be factored as \( 6x(x + 2) \).
- \( 4x + 12 \) can be factored as \( 4(x + 3) \).

So the expression becomes:

\[
\frac{3(x + 2)}{(x - 3)(x + 3)} \div \frac{6x(x + 2)}{4(x + 3)}
\]

### Step 2: Convert division into multiplication.
When dividing by a fraction, we multiply by its reciprocal:

\[
\frac{3(x + 2)}{(x - 3)(x + 3)} \times \frac{4(x + 3)}{6x(x + 2)}
\]

### Step 3: Simplify the expression.
- Cancel out \( (x + 2) \) from both the numerator and denominator.
- Cancel out \( (x + 3) \) from both the numerator and denominator.
- Simplify the constant terms: \( \frac{3 \times 4}{6x} = \frac{12}{6x} = \frac{2}{x} \).

So, the simplified expression is:

\[
\frac{2}{x - 3}
\]

This is the final simplified form.

Would you like further details on this or have any questions?

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Here are 5 related questions to expand on the solution:

1. How would you factor more complex quadratic expressions?
2. What other methods can be used to simplify rational expressions?
3. What happens if \(x = 3\) in the original expression?
4. How does the concept of reciprocal division work for algebraic fractions?
5. How do you handle more complicated division of rational expressions?

**Tip:** Always check for common factors to cancel in both the numerator and the denominator before performing operations.

Simplify the expression \( \frac{3x + 6}{x^2 - 9} \div \frac{6x^2 + 12x}{4x + 12} \).

Learn how to simplify the rational expression \( \frac{3x + 6}{x^2 - 9} \div \frac{6x^2 + 12x}{4x + 12} \) by using factorization techniques and understanding the division of rational expressions. This problem includes factoring quadratic expressions, the difference of squares, and reciprocal division, suitable for students in Grades 9-11.

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