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To solve \((4 + 5i)(8 + 7i)\), we use the distributive property (FOIL method) for multiplying complex numbers. Let's walk through the steps:

1. **Multiply the real parts**:  
   \(4 \times 8 = 32\)

2. **Multiply the outer terms**:  
   \(4 \times 7i = 28i\)

3. **Multiply the inner terms**:  
   \(5i \times 8 = 40i\)

4. **Multiply the imaginary parts**:  
   \(5i \times 7i = 35i^2\)

Since \(i^2 = -1\), \(35i^2 = 35 \times (-1) = -35\).

Now, sum all the parts:
- Real part: \(32 - 35 = -3\)
- Imaginary part: \(28i + 40i = 68i\)

Thus, the simplified expression is:  
\[
(4 + 5i)(8 + 7i) = -3 + 68i
\]

Would you like more details or have any other questions? Here are some related questions:

1. How do you multiply two complex numbers?
2. What is the significance of \(i^2 = -1\) in complex number calculations?
3. How can we generalize this method for any complex number multiplication?
4. What are the geometric interpretations of multiplying complex numbers?
5. Can the result of multiplying two complex numbers ever be a purely real number?

**Tip:** When multiplying complex numbers, always remember to apply \(i^2 = -1\) to simplify the result.

Learn how to multiply complex numbers using the distributive property. This example simplifies (4 + 5i)(8 + 7i) to -3 + 68i by applying key concepts in algebra and imaginary numbers. Suitable for Grades 10-12.

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