The expression you've provided is:

\[
\frac{4 - i}{3 + 2i}
\]

To simplify this expression, we can multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of \(3 + 2i\) is \(3 - 2i\).

So, let's do that:

\[
\frac{(4 - i) \cdot (3 - 2i)}{(3 + 2i) \cdot (3 - 2i)}
\]

### Step 1: Multiply the numerator
\[
(4 - i)(3 - 2i) = 4 \cdot 3 - 4 \cdot 2i - i \cdot 3 + i \cdot 2i
\]
\[
= 12 - 8i - 3i - 2i^2
\]

Since \(i^2 = -1\), the expression simplifies to:
\[
= 12 - 8i - 3i + 2 = 14 - 11i
\]

### Step 2: Multiply the denominator
\[
(3 + 2i)(3 - 2i) = 3^2 - (2i)^2 = 9 - 4i^2
\]
\[
= 9 - 4(-1) = 9 + 4 = 13
\]

### Final Answer:
\[
\frac{4 - i}{3 + 2i} = \frac{14 - 11i}{13}
\]

This can be separated into real and imaginary parts:
\[
\frac{14}{13} - \frac{11}{13}i
\]

So, the simplified form of \(\frac{4 - i}{3 + 2i}\) is:
\[
\frac{14}{13} - \frac{11}{13}i
\]

Would you like to see more details or have any other questions?

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Here are 8 related questions:

1. How do you find the conjugate of a complex number?
2. What is the significance of multiplying by the conjugate in complex number division?
3. Can you simplify \(\frac{2 + 3i}{1 - 4i}\) using the same method?
4. What happens if you multiply two complex conjugates together?
5. How do you add and subtract complex numbers?
6. How do you multiply two complex numbers without using their conjugates?
7. What is the modulus of a complex number?
8. How do you find the argument of a complex number?

**Tip:** When working with complex numbers, always remember that \(i^2 = -1\). This fundamental property simplifies many expressions involving imaginary numbers.

Learn how to simplify the division of complex numbers (4 - i) / (3 + 2i) step-by-step with detailed explanations and examples. This problem involves using the conjugate of a complex number in division, illustrating advanced concepts in complex number arithmetic. Suitable for advanced high school students and learners interested in mathematics.

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