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To simplify \(\frac{-8 + 6i}{4 + 2i}\) in the form \(a + bi\), we should multiply both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of \(4 + 2i\) is \(4 - 2i\).

### Step-by-Step Solution:

1. **Set up the expression:**
   \[
   \frac{-8 + 6i}{4 + 2i} \times \frac{4 - 2i}{4 - 2i}
   \]

2. **Multiply the numerators:**
   \[
   (-8 + 6i)(4 - 2i)
   \]
   Expanding this:
   \[
   = (-8)(4) + (-8)(-2i) + (6i)(4) + (6i)(-2i)
   \]
   \[
   = -32 + 16i + 24i - 12i^2
   \]
   Since \(i^2 = -1\), substitute \(-1\) for \(i^2\):
   \[
   = -32 + 16i + 24i + 12
   \]
   \[
   = -20 + 40i
   \]

3. **Multiply the denominators:**
   \[
   (4 + 2i)(4 - 2i)
   \]
   Using the difference of squares formula:
   \[
   = 4^2 - (2i)^2
   \]
   \[
   = 16 - 4(-1)
   \]
   \[
   = 16 + 4 = 20
   \]

4. **Combine the results:**
   \[
   \frac{-20 + 40i}{20} = \frac{-20}{20} + \frac{40i}{20}
   \]
   Simplify each term:
   \[
   = -1 + 2i
   \]

### Final Answer:
\[
-1 + 2i
\]

Would you like more details on this solution or have any questions?

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### Related Questions:
1. How do you find the complex conjugate of a complex number?
2. What is the significance of multiplying by the complex conjugate in division of complex numbers?
3. How would the solution change if the denominator was \(4 - 2i\) instead?
4. How can we convert complex numbers from rectangular form to polar form?
5. What is the geometric interpretation of dividing two complex numbers?

### Tip:
When dividing complex numbers, always multiply by the conjugate of the denominator to simplify the expression.

Write the quotient in the form a + bi: (-8 + 6i) / (4 + 2i).

Learn how to divide complex numbers with this step-by-step solution for (-8 + 6i) / (4 + 2i). This problem covers essential concepts in complex number division, including using the complex conjugate. Suitable for high school students.

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