Let's solve the multiplication of two complex numbers, \((1 - 4i)\) and \((-4 + 2i)\).

We use the distributive property (also known as the FOIL method) for multiplying complex numbers:

\[
(a + bi)(c + di) = (ac) + (adi) + (bci) + (bdi^2)
\]

Here, \(a = 1\), \(b = -4\), \(c = -4\), and \(d = 2\).

Now, let's calculate each part:

1. Multiply the real parts: 
   \[
   1 \cdot -4 = -4
   \]
2. Multiply the real part of the first and imaginary part of the second:
   \[
   1 \cdot 2i = 2i
   \]
3. Multiply the imaginary part of the first and real part of the second:
   \[
   -4i \cdot -4 = 16i
   \]
4. Multiply the imaginary parts:
   \[
   -4i \cdot 2i = -8i^2 = 8 \quad (\text{since } i^2 = -1)
   \]

Now, let's combine all the terms:
\[
-4 + 2i + 16i + 8
\]

Combine the real parts and imaginary parts separately:
\[
(-4 + 8) + (2i + 16i) = 4 + 18i
\]

Thus, the result of multiplying \((1 - 4i)\) and \((-4 + 2i)\) is:
\[
\boxed{4 + 18i}
\]

Would you like further details on any step? Here are some related questions you can explore:

1. How do you add or subtract complex numbers?
2. What is the geometric interpretation of complex numbers?
3. How do you find the modulus of a complex number?
4. Can you explain how to divide complex numbers?
5. What are the applications of complex numbers in engineering?

**Tip:** Always remember that \(i^2 = -1\), which is crucial for simplifying complex number multiplications.

Learn how to multiply complex numbers, using the distributive property (FOIL method) to solve (1-4i) · (-4+2i). This detailed solution includes key algebraic concepts and demonstrates how to simplify complex number expressions. Ideal for high school students studying Algebra or Complex Numbers.

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