To solve the product of the two complex numbers \((12 + 9i)\) and \((-9 - 3i)\), we can use the distributive property (also known as the FOIL method when dealing with binomials).

\[
(12 + 9i)(-9 - 3i)
\]

Expanding this:

\[
= 12(-9) + 12(-3i) + 9i(-9) + 9i(-3i)
\]

Simplify each term:

\[
= -108 - 36i - 81i - 27i^2
\]

Recall that \(i^2 = -1\), so:

\[
= -108 - 36i - 81i + 27
\]

Combine like terms:

\[
= (-108 + 27) + (-36i - 81i)
\]

\[
= -81 - 117i
\]

So, the product of \((12 + 9i)\) and \((-9 - 3i)\) is:

\[
\boxed{-81 - 117i}
\]

Would you like further details or have any questions? Here are five related questions you might find interesting:

1. What happens when you multiply two complex conjugates?
2. How do you find the magnitude of a complex number?
3. What is the polar form of a complex number?
4. How can complex numbers be represented graphically?
5. What is the significance of the imaginary unit \(i\) in complex numbers?

**Tip:** When multiplying complex numbers, always remember to apply the distributive property carefully and simplify using the fact that \(i^2 = -1\).

Discover how to multiply the complex numbers (12 + 9i) and (-9 - 3i) using the distributive property and simplifying using i^2 = -1. This step-by-step guide covers key concepts in complex number multiplication, making it ideal for high school students in Grades 9-12.

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