The problem asks you to simplify the following expression:

$-\left( \frac{1 + 5i}{7} \right) + \frac{1 + i}{7}$ where $i$ represents the imaginary unit.

Step-by-step simplification:

1. First, distribute the negative sign across the first fraction: $-\left( \frac{1 + 5i}{7} \right) = \frac{-1 - 5i}{7}$

2. Now rewrite the entire expression: $\frac{-1 - 5i}{7} + \frac{1 + i}{7}$

3. Since both fractions have the same denominator, we can combine the numerators: $\frac{-1 - 5i + 1 + i}{7} = \frac{(-1 + 1) + (-5i + i)}{7}$

4. Simplify the numerators: $0 - 4i = -4i$

5. Now the expression becomes: $\frac{-4i}{7}$

This is the simplified form of the expression. Since the problem asks for the answer in the form $a + bi$, we can write it as:

$0 - \frac{4i}{7}$ or simply:

$0 - \frac{4}{7}i$

Thus, the answer is: $0 - \frac{4}{7}i$

Do you need any further details or have any questions?

Here are 5 related questions:

1. How do you simplify complex fractions involving real and imaginary parts?
2. Can you explain the steps to add or subtract two complex numbers?
3. What is the significance of the imaginary unit $i$ in complex numbers?
4. How would you represent a complex number graphically?
5. Can complex numbers be used in real-world applications? If so, how?

Tip:

Always remember that when simplifying complex fractions, you can combine numerators directly if the denominators are the same.