https://file.aiquickdraw.com/mathbot/file/1725627349176oq6ifm8t.jpg

The expression you need to simplify is:

\[
\left( \frac{2n}{8n + 4} \right) \left( \frac{3n + 2}{3n - 2} \right)
\]

Let's go step by step:

1. **Factor the denominators (if possible)**:
   - The first denominator \( 8n + 4 \) can be factored as:
     \[
     8n + 4 = 4(2n + 1)
     \]
   - The second denominator \( 3n - 2 \) is already simplified.

So, the expression becomes:

\[
\left( \frac{2n}{4(2n + 1)} \right) \left( \frac{3n + 2}{3n - 2} \right)
\]

2. **Multiply the numerators and denominators**:
   - Numerator: \( 2n(3n + 2) = 6n^2 + 4n \)
   - Denominator: \( 4(2n + 1)(3n - 2) \)

Thus, the expression becomes:

\[
\frac{6n^2 + 4n}{4(2n + 1)(3n - 2)}
\]

3. **Extract the simplified numerator**:
   The numerator of the simplified expression is:
   \[
   6n^2 + 4n
   \]

Would you like more details or have any further questions?

Here are some related questions for you:
1. How do you factor polynomial expressions?
2. What is the process of multiplying rational expressions?
3. How can you check if the numerator or denominator has common factors?
4. What are some ways to simplify complex fractions?
5. Can you factor any further after simplification?

**Tip**: When simplifying expressions, always look for common factors in the numerator and denominator to reduce the expression further.

Simplify the expression (2n / (8n + 4)) * ((3n + 2) / (3n - 2)) and find the numerator of the simplified expression.

Learn how to simplify the rational expression (2n / (8n + 4)) * ((3n + 2) / (3n - 2)). This problem covers essential algebra concepts such as multiplying rational expressions and factoring. Ideal for students in Grades 9-12.

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