The problem asks you to simplify the expression:

$\frac{1 + 6i}{2 - 4i}$

Steps to Simplify:

1. Multiply both numerator and denominator by the conjugate of the denominator. The conjugate of $2 - 4i$ is $2 + 4i$.

   Multiply both the numerator and the denominator by $2 + 4i$: $\frac{(1 + 6i)(2 + 4i)}{(2 - 4i)(2 + 4i)}$

2. Simplify the denominator: $(2 - 4i)(2 + 4i) = 2^2 - (4i)^2 = 4 - (-16) = 4 + 16 = 20$ So, the denominator is 20.

3. Simplify the numerator: $(1 + 6i)(2 + 4i) = 1 \cdot 2 + 1 \cdot 4i + 6i \cdot 2 + 6i \cdot 4i$ $= 2 + 4i + 12i + 24i^2$ Recall that $i^2 = -1$, so: $24i^2 = 24 \times (-1) = -24$ Therefore, the numerator becomes: $2 + 4i + 12i - 24 = -22 + 16i$

4. Final expression: Now divide both the real and imaginary parts by 20: $\frac{-22 + 16i}{20} = \frac{-22}{20} + \frac{16i}{20} = -\frac{11}{10} + \frac{4i}{5}$

So, the simplified expression is:

$-\frac{11}{10} + \frac{4i}{5}$

Would you like further details on any part of this solution?

Here are some related questions:

1. How do you find the conjugate of a complex number?
2. What is the significance of multiplying by the conjugate?
3. Why does $i^2$ equal -1?
4. How would the process differ if the denominator had been a purely real number?
5. How can complex fractions be simplified differently?

Tip: Always remember to express your final answer in the standard form $a + bi$, where $a$ and $b$ are real numbers.