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?

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.