Let's break down and solve the expression step by step:

$\frac{2}{3} \times \left(-\frac{5}{7}\right) + \frac{7}{3} + \frac{2}{3} \times \left(-\frac{2}{7}\right)$

Step 1: Multiply the fractions.

First, we'll perform the multiplications:

$\frac{2}{3} \times \left(-\frac{5}{7}\right) = -\frac{10}{21}$ $\frac{2}{3} \times \left(-\frac{2}{7}\right) = -\frac{4}{21}$

Step 2: Rewrite the expression.

Now, rewrite the expression with the results from the multiplications:

$-\frac{10}{21} + \frac{7}{3} - \frac{4}{21}$

Step 3: Find a common denominator.

The common denominator between 21 and 3 is 21. We'll rewrite $\frac{7}{3}$ with a denominator of 21:

$\frac{7}{3} = \frac{49}{21}$

Now the expression becomes:

$-\frac{10}{21} + \frac{49}{21} - \frac{4}{21}$

Step 4: Combine the terms.

Since the denominators are the same, we can combine the numerators:

$\frac{-10 + 49 - 4}{21} = \frac{35}{21}$

Step 5: Simplify the fraction.

$\frac{35}{21}$ simplifies to:

$\frac{5}{3}$

Final Answer:

The simplified value of the expression is:

$\frac{5}{3}$

Would you like further details or explanations on any part of the solution?

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Here are 5 related questions to further your understanding:

1. How do you find a common denominator between two fractions?
2. Can you explain how to simplify a fraction?
3. What are the rules for multiplying fractions?
4. How do you add and subtract fractions with different denominators?
5. Can you provide another example of simplifying a complex fraction?

Tip: When working with fractions, always look for opportunities to simplify as early as possible to make calculations easier.