To solve the expression $\frac{2}{5} \times \left( -\frac{2}{3} + \frac{3}{4} \right)$, follow these steps:

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Step 1: Simplify the parentheses

Combine $-\frac{2}{3} + \frac{3}{4}$ by finding a common denominator.

The denominators are $3$ and $4$, so the least common denominator (LCD) is $12$. Rewrite each fraction: $-\frac{2}{3} = -\frac{8}{12}, \quad \frac{3}{4} = \frac{9}{12}.$

Now, add the fractions: $-\frac{8}{12} + \frac{9}{12} = \frac{-8 + 9}{12} = \frac{1}{12}.$

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Step 2: Multiply by $\frac{2}{5}$

Now substitute $\frac{1}{12}$ into the expression: $\frac{2}{5} \times \frac{1}{12}.$

Multiply the numerators and denominators: $\frac{2 \cdot 1}{5 \cdot 12} = \frac{2}{60}.$

Simplify $\frac{2}{60}$: $\frac{2}{60} = \frac{1}{30}.$

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Final Answer:

$\boxed{\frac{1}{30}}$

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Would you like further explanation or have additional questions? Here are some related questions you might explore:

1. How do you find the least common denominator (LCD)?
2. What are the rules for multiplying fractions?
3. How do you handle negative fractions in addition or subtraction?
4. What is the process for simplifying fractions?
5. Can you explain how to distribute fractions over parentheses?

Tip: Always simplify the fractions at each step to make calculations easier.