The expression you provided involves dividing two complex numbers:

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

To solve this, we need to multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of $6 - i$ is $6 + i$.

Step-by-step solution:

1. Multiply numerator and denominator by the conjugate:

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

2. Simplify the denominator:

$(6 - i)(6 + i) = 6^2 - i^2 = 36 - (-1) = 36 + 1 = 37$

So, the denominator simplifies to 37.

3. Multiply the numerators:

$(4 + 2i)(6 + i) = 4(6) + 4(i) + 2i(6) + 2i(i)$ $= 24 + 4i + 12i + 2i^2$ $= 24 + 16i + 2(-1)$ $= 24 + 16i - 2 = 22 + 16i$

4. Combine the results:

$\frac{22 + 16i}{37}$

This can be written as two separate terms:

$\frac{22}{37} + \frac{16}{37}i$

Thus, the quotient is:

$\frac{22}{37} + \frac{16}{37}i$

Final Answer:

$\boxed{\frac{22}{37} + \frac{16}{37}i}$

Would you like a more detailed explanation of any step?

Here are some related questions to explore this further:

1. How do you find the conjugate of a complex number?
2. Why do we multiply by the conjugate when dividing complex numbers?
3. Can you simplify complex numbers in polar form?
4. What happens if you multiply two complex conjugates?
5. What are some applications of complex number division in physics or engineering?

Tip: When working with complex numbers, always simplify using the conjugate to avoid imaginary terms in the denominator.