To solve \(\frac{4 - 2i}{1 + 2i}\), we multiply both the numerator and the denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator.

### Step 1: Multiply by the conjugate of the denominator

The conjugate of \(1 + 2i\) is \(1 - 2i\). Multiply both the numerator and denominator by this conjugate:
\[
\frac{(4 - 2i)(1 - 2i)}{(1 + 2i)(1 - 2i)}
\]

### Step 2: Simplify the denominator

The denominator is a difference of squares:
\[
(1 + 2i)(1 - 2i) = 1^2 - (2i)^2 = 1 - 4(-1) = 1 + 4 = 5
\]
So the denominator becomes 5.

### Step 3: Expand the numerator

Now expand the numerator:
\[
(4 - 2i)(1 - 2i) = 4(1) + 4(-2i) - 2i(1) - 2i(-2i)
\]
\[
= 4 - 8i - 2i + 4i^2
\]
Since \(i^2 = -1\), we get:
\[
= 4 - 8i - 2i + 4(-1) = 4 - 8i - 2i - 4 = -8 - 10i
\]

### Step 4: Divide by the denominator

Now divide the result by the denominator, which is 5:
\[
\frac{-8 - 10i}{5} = \frac{-8}{5} + \frac{-10i}{5} = -\frac{8}{5} - 2i
\]

### Final Answer:
\[
\frac{4 - 2i}{1 + 2i} = -\frac{8}{5} - 2i
\]

Would you like further explanation or have any questions?

Here are some related questions to explore:
1. What is the geometric interpretation of complex numbers?
2. How do you find the magnitude of a complex number?
3. What is the significance of multiplying by the conjugate in complex number division?
4. How can you express the result in polar form?
5. What happens when you divide complex numbers using trigonometric identities?

**Tip**: Multiplying by the conjugate is a key technique for eliminating imaginary parts from denominators.

This detailed solution demonstrates how to solve (4 - 2i) / (1 + 2i) by multiplying by the conjugate of the denominator, applying key concepts in complex number arithmetic. Learn about the difference of squares and simplifying complex expressions. Ideal for Grades 10-12 students.

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