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The question asks to express the sum of two fractions as a single fraction:

\[
\frac{3}{3x + 1} + \frac{2}{4x - 2}
\]

### Step-by-step solution:

1. **Find the least common denominator (LCD):**

   The denominators are \(3x + 1\) and \(4x - 2\). To add fractions, we need to express them with a common denominator.

   The LCD is simply the product of the two denominators, since they are different:
   \[
   \text{LCD} = (3x + 1)(4x - 2)
   \]

2. **Rewrite each fraction with the LCD as the denominator:**

   - For the first fraction \(\frac{3}{3x + 1}\), multiply both the numerator and denominator by \(4x - 2\):
     \[
     \frac{3}{3x + 1} = \frac{3(4x - 2)}{(3x + 1)(4x - 2)} = \frac{12x - 6}{(3x + 1)(4x - 2)}
     \]
   
   - For the second fraction \(\frac{2}{4x - 2}\), multiply both the numerator and denominator by \(3x + 1\):
     \[
     \frac{2}{4x - 2} = \frac{2(3x + 1)}{(3x + 1)(4x - 2)} = \frac{6x + 2}{(3x + 1)(4x - 2)}
     \]

3. **Add the two fractions:**
   Now that both fractions have the same denominator, we can add them directly:
   \[
   \frac{12x - 6}{(3x + 1)(4x - 2)} + \frac{6x + 2}{(3x + 1)(4x - 2)} = \frac{(12x - 6) + (6x + 2)}{(3x + 1)(4x - 2)}
   \]
   
   Simplify the numerator:
   \[
   12x - 6 + 6x + 2 = 18x - 4
   \]

4. **Final result:**
   \[
   \frac{18x - 4}{(3x + 1)(4x - 2)}
   \]

Thus, the sum of the two fractions expressed as a single fraction is:

\[
\frac{18x - 4}{(3x + 1)(4x - 2)}
\]

Would you like further details or clarification on any steps?

### Related Questions:
1. How do you simplify complex fractions?
2. What is the importance of finding the least common denominator in fraction operations?
3. Can you express \(\frac{18x - 4}{(3x + 1)(4x - 2)}\) in a factored form?
4. How do you handle cases when the denominators of two fractions share common factors?
5. How would you solve \(\frac{3}{3x + 1} = \frac{2}{4x - 2}\)?

### Tip:
When adding fractions with different denominators, always find the least common denominator to simplify the process efficiently.

Express \(\frac{3}{3x + 1} + \frac{2}{4x - 2}\) as a single fraction.

Learn how to express \(\frac{3}{3x + 1} + \frac{2}{4x - 2}\) as a single fraction by finding the least common denominator (LCD) and simplifying the expression. Suitable for students in Grades 9-11.

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