The problem asks to simplify the expression and write the answer in the form $a + bi$. The expression given is:

$-\left(\frac{1 + 9i}{4}\right) + \frac{7}{3} - \frac{10 - 2i}{3}$

Step 1: Simplify Each Term

1. First Term: $-\left(\frac{1 + 9i}{4}\right) = -\frac{1 + 9i}{4} = -\frac{1}{4} - \frac{9i}{4}$

2. Second Term: $\frac{7}{3}$ This term is already in its simplest form.

3. Third Term: $\frac{10 - 2i}{3} = \frac{10}{3} - \frac{2i}{3}$

Step 2: Combine the Real Parts

Add the real parts together: $-\frac{1}{4} + \frac{7}{3} - \frac{10}{3}$

To add these fractions, convert them to have a common denominator:

$-\frac{1}{4} + \frac{7}{3} - \frac{10}{3} = -\frac{1}{4} + \frac{7 - 10}{3} = -\frac{1}{4} - \frac{3}{3} = -\frac{1}{4} - 1$

Now, combine $-\frac{1}{4}$ and $-1$:

$-\frac{1}{4} - \frac{4}{4} = -\frac{5}{4}$

Step 3: Combine the Imaginary Parts

Now, add the imaginary parts: $-\frac{9i}{4} - \frac{2i}{3}$

To add these fractions, convert them to have a common denominator:

$-\frac{9i}{4} - \frac{2i}{3} = -\frac{27i}{12} - \frac{8i}{12} = -\frac{35i}{12}$

Final Answer

So the expression simplifies to:

$-\frac{5}{4} - \frac{35i}{12}$

This is already in the form $a + bi$ where $a = -\frac{5}{4}$ and $b = -\frac{35}{12}$.

Would you like any additional details or have any questions?

Here are some related questions:

1. How do you multiply complex numbers?
2. What are the different ways to represent complex numbers?
3. How do you find the magnitude of a complex number?
4. What is the polar form of a complex number?
5. How do you divide complex numbers?

Tip: When working with complex numbers, always separate the real and imaginary parts to simplify the calculations.