**Multiply and simplify the following complex numbers:**

\[(\pink{1-5i}) \cdot (\blue{3-2i})\]

To multiply and simplify the given complex numbers \((1 - 5i) \cdot (3 - 2i)\), we will use the distributive property, often referred to as the FOIL (First, Outer, Inner, Last) method when dealing with binomials.

Let's proceed step-by-step:

1. **First**: Multiply the first terms from each binomial:
   \[
   1 \cdot 3 = 3
   \]

2. **Outer**: Multiply the outer terms:
   \[
   1 \cdot (-2i) = -2i
   \]

3. **Inner**: Multiply the inner terms:
   \[
   -5i \cdot 3 = -15i
   \]

4. **Last**: Multiply the last terms:
   \[
   -5i \cdot (-2i) = 10i^2
   \]

Next, we simplify by combining like terms. But first, recall that \(i^2 = -1\), so:
\[
10i^2 = 10(-1) = -10
\]

Now, let's add up all the terms:
\[
3 - 2i - 15i - 10
\]

Combine the real and imaginary parts:
\[
(3 - 10) + (-2i - 15i) = -7 - 17i
\]

Thus, the simplified result of multiplying \((1 - 5i) \cdot (3 - 2i)\) is:
\[
\boxed{-7 - 17i}
\]

Would you like to see further details or have any questions?

Here are 5 related questions:
1. How do you add and subtract complex numbers?
2. What is the conjugate of a complex number, and how is it used in division?
3. How do you find the modulus of a complex number?
4. What is the polar form of a complex number?
5. How do you multiply complex numbers in polar form?

**Tip:** Always remember that \(i^2 = -1\). This is crucial for simplifying expressions involving complex numbers.

Learn how to multiply and simplify complex numbers (1 - 5i) * (3 - 2i) using the FOIL method and properties of the imaginary unit (i^2 = -1). Follow a step-by-step solution to combine real and imaginary parts effectively. Suitable for advanced high school students studying complex numbers.

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