To divide the complex numbers $\frac{6 - 2i}{4 + i}$, follow these steps:

Step 1: Multiply by the Conjugate

Multiply the numerator and denominator by the conjugate of the denominator $4 - i$:

$\frac{(6 - 2i)(4 - i)}{(4 + i)(4 - i)}$

Step 2: Expand the Denominator

Using the difference of squares formula:

$(4 + i)(4 - i) = 4^2 - i^2 = 16 - (-1) = 17$

Step 3: Expand the Numerator

Using the distributive property:

$(6 - 2i)(4 - i) = 6(4) + 6(-i) + (-2i)(4) + (-2i)(-i)$

$= 24 - 6i - 8i + 2i^2$

Since $i^2 = -1$, we get:

$= 24 - 14i + 2(-1)$

$= 24 - 14i - 2$

$= 22 - 14i$

Step 4: Write the Final Answer

$\frac{22 - 14i}{17} = \frac{22}{17} - \frac{14}{17}i$

Thus, the correct answer is:

$\boxed{A.) \frac{22}{17} - \frac{14}{17}i}$

---

Would you like a step-by-step breakdown of any part?

Related Questions:

1. How do you find the conjugate of a complex number?
2. Why does multiplying by the conjugate remove the imaginary part in the denominator?
3. What is the general formula for dividing complex numbers?
4. How do you represent complex numbers in polar form?
5. How does complex number division relate to vector rotation?

Tip:

Always multiply the numerator and denominator by the conjugate when dividing complex numbers to eliminate the imaginary part from the denominator.