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The math problem presents a complex fraction:

\[
\frac{\frac{6x + 12}{x}}{\frac{3x + 9}{5x}}
\]

Let's simplify this expression step by step.

### Step 1: Simplify each numerator and denominator.
First, factor common terms:

\[
6x + 12 = 6(x + 2)
\quad \text{and} \quad 3x + 9 = 3(x + 3)
\]

Thus, the expression becomes:

\[
\frac{\frac{6(x + 2)}{x}}{\frac{3(x + 3)}{5x}}
\]

### Step 2: Simplify the complex fraction.
To divide fractions, multiply the numerator by the reciprocal of the denominator:

\[
\frac{6(x + 2)}{x} \times \frac{5x}{3(x + 3)} = \frac{6(x + 2) \cdot 5x}{x \cdot 3(x + 3)}
\]

### Step 3: Cancel out common terms.
The \(x\) in the numerator and denominator cancels out:

\[
\frac{6 \cdot 5(x + 2)}{3(x + 3)} = \frac{30(x + 2)}{3(x + 3)}
\]

Now, simplify the coefficient:

\[
\frac{30(x + 2)}{3(x + 3)} = \frac{10(x + 2)}{x + 3}
\]

### Final Answer:
\[
\frac{10(x + 2)}{x + 3}
\]

Let me know if you want more details or have any questions!

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Here are 5 questions to explore this topic further:
1. What is the general method to simplify complex fractions?
2. How does factoring help in simplifying algebraic expressions?
3. What is the importance of recognizing common factors in algebraic fractions?
4. How would the solution change if the numerator and denominator had different constants?
5. How can you check if your final simplified answer is correct?

**Tip**: Always look for common factors to cancel in fractions before multiplying or dividing them!

Simplify the complex fraction: (6x + 12)/x ÷ (3x + 9)/(5x)

Learn how to simplify the complex fraction (6x + 12)/x ÷ (3x + 9)/(5x). This step-by-step solution involves factoring and the division of algebraic fractions, suitable for students in Grades 8-10.

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