To multiply the two complex numbers \( (2 + 3i) \) and \( (1 - 4i) \), we will use the distributive property (also known as the FOIL method).

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

Now, calculate each term:

\[
= 2 - 8i + 3i - 12i^2
\]

Since \( i^2 = -1 \), substitute that in:

\[
= 2 - 8i + 3i - 12(-1)
\]

\[
= 2 - 8i + 3i + 12
\]

Combine like terms:

\[
= (2 + 12) + (-8i + 3i)
\]

\[
= 14 - 5i
\]

So, the product of \( (2 + 3i)(1 - 4i) \) is:

\[
\boxed{14 - 5i}
\]

Would you like a more detailed explanation or have any questions?

Here are 5 related questions to further expand your understanding:
1. How do you multiply complex numbers using the FOIL method?
2. What happens when you square a complex number like \( (a + bi)^2 \)?
3. What is the significance of \( i^2 = -1 \) in complex numbers?
4. How do you find the conjugate of a complex number?
5. How can you represent complex numbers on a coordinate plane?

**Tip:** When multiplying complex numbers, always remember that \( i^2 = -1 \), which turns any imaginary square into a real negative number!

Learn how to multiply the complex numbers (2 + 3i) and (1 - 4i) using the distributive property and properties of the imaginary unit. Detailed step-by-step explanation suitable for Grades 10-12.

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