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The given expression is \((3 + 4i)(2 - 2i)\), where \(i\) represents the imaginary unit.

Let's multiply these two complex numbers using the distributive property (FOIL method):

\[
(3 + 4i)(2 - 2i)
\]

### Step-by-step multiplication:
1. First, multiply the first terms:
   \[
   3 \times 2 = 6
   \]
2. Multiply the outer terms:
   \[
   3 \times -2i = -6i
   \]
3. Multiply the inner terms:
   \[
   4i \times 2 = 8i
   \]
4. Multiply the last terms:
   \[
   4i \times -2i = -8i^2
   \]
   Since \(i^2 = -1\), this becomes:
   \[
   -8i^2 = -8(-1) = 8
   \]

### Now, combine all these results:
\[
6 - 6i + 8i + 8
\]

Combine like terms:
\[
(6 + 8) + (-6i + 8i) = 14 + 2i
\]

### Final result:
\[
(3 + 4i)(2 - 2i) = 14 + 2i
\]

Would you like more details on any part of this? Here are five related questions for further exploration:

1. What is the general formula for multiplying two complex numbers?
2. How does \(i^2 = -1\) affect complex number calculations?
3. Can you represent the result \(14 + 2i\) on the complex plane?
4. What is the geometric interpretation of multiplying complex numbers?
5. How would you divide two complex numbers?

**Tip:** Always remember that when multiplying two complex numbers, use the FOIL method and simplify using the fact that \(i^2 = -1\).

Learn how to multiply two complex numbers (3 + 4i) and (2 - 2i) using the FOIL method. This example covers key concepts in algebra and complex number theory, suitable for high school students (Grades 10-12).

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