To multiply the two complex numbers \((1 - 5i)\) and \((3 - 2i)\), we use the distributive property (FOIL method) to expand the product:

\[
(1 - 5i) \cdot (3 - 2i) = 1 \cdot 3 + 1 \cdot (-2i) - 5i \cdot 3 - 5i \cdot (-2i)
\]

Now, let's calculate each term:

1. \(1 \cdot 3 = 3\)
2. \(1 \cdot (-2i) = -2i\)
3. \(-5i \cdot 3 = -15i\)
4. \(-5i \cdot (-2i) = 10i^2\)

Since \(i^2 = -1\), we substitute \(i^2\) with \(-1\):

\[
10i^2 = 10 \times (-1) = -10
\]

Combining all the terms gives:

\[
3 - 2i - 15i - 10
\]

Now, let's combine like terms:

\[
(3 - 10) + (-2i - 15i) = -7 - 17i
\]

So, the product of the two complex numbers is:

\[
\boxed{-7 - 17i}
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you multiply two complex numbers in polar form?
2. What is the geometric interpretation of multiplying two complex numbers?
3. Can you find the magnitude and argument of a complex number?
4. How do you add and subtract complex numbers?
5. What is the conjugate of a complex number, and why is it useful?

### Tip:
Remember that multiplying complex numbers involves both the real and imaginary parts. Using \(i^2 = -1\) is crucial for simplifying your results!

Explore how to multiply the complex numbers (1 - 5i) and (3 - 2i) using the distributive property (FOIL method). This step-by-step solution breaks down the process, including key concepts like the imaginary unit (i^2 = -1) and combining like terms. Suitable for high school students (Grades 10-12).

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