To solve this problem, we need to simplify each expression in the rows and match them with the equivalent expressions in the columns.

1. Row 1: $\left(\frac{1}{3}x + \frac{2}{3}\right) + \left(\frac{2}{3}x - \frac{1}{3}\right)$

   • Simplify each term: Combine like terms for $x$ and constants.
   • $\frac{1}{3}x + \frac{2}{3}x = x$ and $\frac{2}{3} - \frac{1}{3} = \frac{1}{3}$.
   • Result: $x + \frac{1}{3}$, which matches $\frac{1}{3}(3x + 1)$.

2. Row 2: $\left(\frac{2}{3}x + \frac{2}{3}\right) - \left(\frac{1}{3}x - \frac{1}{3}\right)$

   • Distribute and combine terms: $\frac{2}{3}x - \frac{1}{3}x = \frac{1}{3}x$ and $\frac{2}{3} + \frac{1}{3} = 1$.
   • Result: $\frac{1}{3}(x + 3)$.

3. Row 3: $\frac{4}{3}x - \frac{2}{3} + \frac{2}{3}x$

   • Combine terms for $x$: $\frac{4}{3}x + \frac{2}{3}x = 2x$.
   • Result: $2x - \frac{2}{3}$, matching $\frac{2}{3}(3x - 1)$.

4. Row 4: $-\frac{4}{3} + \frac{2}{3} - \frac{1}{3}x$

   • Combine constants: $-\frac{4}{3} + \frac{2}{3} = -\frac{2}{3}$.
   • Result: $-\frac{2}{3} - \frac{1}{3}x$, which matches $-\frac{1}{3}(x + 2)$.

Based on the analysis, here are the correct matches:

• Row 1: Select $\frac{1}{3}(3x + 1)$.
• Row 2: Select $\frac{1}{3}(x + 3)$.
• Row 3: Select $\frac{2}{3}(3x - 1)$.
• Row 4: Select $-\frac{1}{3}(x + 2)$.

Would you like a detailed step-by-step breakdown of each row?

Here are five related questions to expand your understanding:

1. How do you simplify expressions with fractional coefficients?
2. What strategies help in combining like terms effectively?
3. How does factoring help in identifying equivalent expressions?
4. Can you verify your answer by expanding each choice to see if they match?
5. What are the benefits of grouping terms in expressions like these?

Tip: Always look to combine like terms and factor out common terms to simplify complex expressions efficiently.