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To perform the indicated operation and simplify, we will use the distributive property (also known as the FOIL method for binomials). The given expression is:

\[ (8 - 12i)(-9 - 10i) \]

Let's expand this step by step:

1. Multiply the first terms:
\[ 8 \times -9 = -72 \]

2. Multiply the outer terms:
\[ 8 \times -10i = -80i \]

3. Multiply the inner terms:
\[ -12i \times -9 = 108i \]

4. Multiply the last terms:
\[ -12i \times -10i = 120i^2 \]

Recall that \( i^2 = -1 \), so:
\[ 120i^2 = 120 \times -1 = -120 \]

Now, let's combine all these results:
\[ -72 - 80i + 108i - 120 \]

Combine like terms (real and imaginary parts separately):
\[ (-72 - 120) + (-80i + 108i) \]
\[ = -192 + 28i \]

So, the simplified form of the given expression as a complex number is:
\[ -192 + 28i \]

Would you like more details or have any questions about this problem?

Here are 5 more questions you might want to ask:

1. How do you multiply complex numbers using the distributive property?
2. Can you explain the FOIL method in the context of complex numbers?
3. What are the properties of imaginary units that help in simplifying complex number operations?
4. How do you add and subtract complex numbers?
5. Can you provide more examples of operations with complex numbers?

**Tip:** When dealing with complex numbers, always remember that \( i^2 = -1 \). This property is crucial for simplifying expressions involving imaginary units.

Learn how to simplify the expression (8 - 12i)(-9 - 10i) step-by-step using the distributive property and the FOIL method for complex numbers. Understand the properties of imaginary units and their role in simplifying complex number operations. Suitable for advanced high school students and anyone interested in complex number arithmetic.

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