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To solve the given expression, let's break it down step by step.

The expression is:

\[
3r^{-1} \cdot \left[ 2(r^{-1} + 2) - r^{-1}\left(2 - r^{-1}\right) - 4 \right]
\]

### Step-by-Step Solution:

1. **Simplify each part inside the brackets**:
   - Start with simplifying the terms inside the brackets:
   \[
   2(r^{-1} + 2) = 2r^{-1} + 4
   \]
   \[
   r^{-1}(2 - r^{-1}) = r^{-1} \cdot 2 - r^{-1} \cdot r^{-1} = 2r^{-1} - r^{-2}
   \]

   Now the expression inside the brackets becomes:
   \[
   (2r^{-1} + 4) - (2r^{-1} - r^{-2}) - 4
   \]

2. **Combine like terms**:
   - First, simplify inside the brackets:
   \[
   2r^{-1} + 4 - 2r^{-1} + r^{-2} - 4
   \]
   Cancel out \(2r^{-1}\) terms, and you get:
   \[
   r^{-2}
   \]

   So, the expression simplifies to:
   \[
   3r^{-1} \cdot r^{-2}
   \]

3. **Multiply the remaining terms**:
   - Multiply the powers of \(r\):
   \[
   3r^{-1} \cdot r^{-2} = 3r^{-3}
   \]

### Final Answer:
\[
3r^{-3}
\]

This is the simplified form of the given expression.

Would you like further clarification on any of the steps or additional details? 

Here are some related questions to deepen your understanding:
1. How do you multiply terms with negative exponents?
2. What is the general rule for combining exponents during multiplication?
3. Can you rewrite the expression \( r^{-3} \) in terms of \( \frac{1}{r^3} \)?
4. What happens if \( r = 0 \) in this expression?
5. How would the solution change if one of the exponents were positive instead of negative?

**Tip:** Always simplify terms inside parentheses first before dealing with multiplication and powers.

How to solve the expression 3r^{-1} * [2(r^{-1} + 2) - r^{-1}(2 - r^{-1}) - 4]

Step-by-step guide to simplify the algebraic expression 3r^{-1} * [2(r^{-1} + 2) - r^{-1}(2 - r^{-1}) - 4]. Learn key concepts like multiplying powers and handling negative exponents. Suitable for high school students in Grades 10-12.

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