To simplify the expression $\frac{6 + 3i}{1 + 5i}$, we'll use the method of multiplying both the numerator and denominator by the complex conjugate of the denominator.

Steps:

1. The complex conjugate of $1 + 5i$ is $1 - 5i$.
2. Multiply both the numerator and the denominator by $1 - 5i$:

$\frac{6 + 3i}{1 + 5i} \times \frac{1 - 5i}{1 - 5i}$

This gives:

$\frac{(6 + 3i)(1 - 5i)}{(1 + 5i)(1 - 5i)}$

3. Now simplify both the numerator and the denominator:

Numerator:

$(6 + 3i)(1 - 5i) = 6(1) + 6(-5i) + 3i(1) + 3i(-5i)$ $= 6 - 30i + 3i - 15i^2$ Since $i^2 = -1$, replace $i^2$ with $-1$: $= 6 - 30i + 3i + 15 = 21 - 27i$

Denominator:

$(1 + 5i)(1 - 5i) = 1^2 - (5i)^2 = 1 - 25i^2$ Since $i^2 = -1$: $= 1 + 25 = 26$

4. Now, the simplified form of the expression is:

$\frac{21 - 27i}{26}$

We can break this down further as:

$\frac{21}{26} - \frac{27}{26}i$

So, the simplified expression is:

$\boxed{\frac{21}{26} - \frac{27}{26}i}$

Would you like any additional details or further clarification on this?

Here are 5 related questions:

1. How do you find the complex conjugate of a complex number?
2. What is the importance of multiplying by the complex conjugate in these cases?
3. How does $i^2 = -1$ affect the simplification of complex expressions?
4. Can you simplify $\frac{4 + 7i}{2 + i}$ using a similar method?
5. What happens if you multiply two purely imaginary numbers?

Tip: When simplifying complex fractions, always multiply by the conjugate of the denominator to eliminate the imaginary part.