Let's simplify the expression step by step:

\[
2i(1 + 3i) - 4i(6 + 3i)
\]

1. **Distribute the first term:**
   \[
   2i(1 + 3i) = 2i \cdot 1 + 2i \cdot 3i = 2i + 6i^2
   \]
   Since \( i^2 = -1 \), we can simplify \( 6i^2 \):
   \[
   6i^2 = 6(-1) = -6
   \]
   Therefore:
   \[
   2i(1 + 3i) = 2i - 6
   \]

2. **Distribute the second term:**
   \[
   -4i(6 + 3i) = -4i \cdot 6 + (-4i) \cdot 3i = -24i - 12i^2
   \]
   Again, using \( i^2 = -1 \):
   \[
   -12i^2 = -12(-1) = 12
   \]
   So:
   \[
   -4i(6 + 3i) = -24i + 12
   \]

3. **Now combine the results:**
   \[
   (2i - 6) + (-24i + 12)
   \]
   Combine the real and imaginary parts:
   \[
   (-6 + 12) + (2i - 24i) = 6 - 22i
   \]

Thus, the simplified expression is:

\[
6 - 22i
\]

Would you like more details on any of the steps?

Here are five related questions:
1. What are the basic properties of imaginary numbers?
2. How do you multiply complex numbers?
3. Can you simplify the expression \( (3 + 4i)(1 - 2i) \)?
4. How do you represent complex numbers graphically?
5. How do you calculate the magnitude of a complex number?

**Tip:** When multiplying by imaginary numbers, always remember that \( i^2 = -1 \), which simplifies the terms.

Learn how to simplify the complex expression 2i(1 + 3i) - 4i(6 + 3i) step by step. This problem explores concepts like the distributive property, imaginary numbers, and key formulas such as i^2 = -1. Suitable for high school students in Grades 9-12.

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