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

The given expression is:

\[
\frac{3 \left( \frac{2}{3} + (-2) \right)}{4} \times \frac{2}{3} \times (-2)^2
\]

### Step 1: Simplify inside the parentheses

Inside the parentheses, you have \(\frac{2}{3} + (-2)\). First, let's convert \(-2\) into a fraction:

\[
\frac{2}{3} + (-2) = \frac{2}{3} + \frac{-6}{3} = \frac{2 - 6}{3} = \frac{-4}{3}
\]

### Step 2: Multiply the result by 3

Now multiply the result of \(\frac{-4}{3}\) by 3:

\[
3 \times \frac{-4}{3} = -4
\]

### Step 3: Divide by 4

Next, divide the result by 4:

\[
\frac{-4}{4} = -1
\]

### Step 4: Multiply by \(\frac{2}{3}\)

Now multiply the result by \(\frac{2}{3}\):

\[
-1 \times \frac{2}{3} = \frac{-2}{3}
\]

### Step 5: Calculate \((-2)^2\)

\((-2)^2 = 4\)

### Step 6: Final multiplication

Now multiply \(\frac{-2}{3}\) by 4:

\[
\frac{-2}{3} \times 4 = \frac{-8}{3}
\]

So, the final result is:

\[
\boxed{\frac{-8}{3}}
\]

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. What happens if you change \((-2)\) to another negative number?
2. How do fractions behave when multiplied by negative numbers?
3. Can you solve the expression if the order of operations changes?
4. What is the impact of raising negative numbers to an odd exponent?
5. How does simplifying fractions in multiplication affect the result?

**Tip**: Always perform operations inside parentheses first to avoid errors with order of operations (PEMDAS).

Step-by-step guide to solving the complex fraction and exponent expression (3(2/3 + -2))/4 x 2/3 x (-2)^2. Learn how to apply order of operations, simplify fractions, and handle negative exponents. Suitable for Grades 8-10 students.

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