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To simplify the expression \(\frac{4 + i}{-5 - 4i}\) into the form \(a + bi\), we need to remove the imaginary unit \(i\) from the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.

### Step-by-Step Solution

1. **Identify the conjugate of the denominator**:
   The denominator is \(-5 - 4i\), so its conjugate is \(-5 + 4i\).

2. **Multiply the numerator and the denominator by the conjugate of the denominator**:
   \[
   \frac{4 + i}{-5 - 4i} \times \frac{-5 + 4i}{-5 + 4i} = \frac{(4 + i)(-5 + 4i)}{(-5 - 4i)(-5 + 4i)}
   \]

3. **Expand the numerator**:
   \[
   (4 + i)(-5 + 4i) = 4 \cdot -5 + 4 \cdot 4i + i \cdot -5 + i \cdot 4i
   \]
   \[
   = -20 + 16i - 5i + 4i^2
   \]
   Since \(i^2 = -1\), we substitute \(4i^2 = 4 \cdot -1 = -4\):
   \[
   = -20 + 16i - 5i - 4 = -24 + 11i
   \]

4. **Simplify the denominator**:
   Using the difference of squares, we get:
   \[
   (-5 - 4i)(-5 + 4i) = (-5)^2 - (4i)^2 = 25 - 16i^2
   \]
   Since \(i^2 = -1\), we substitute \(16i^2 = 16 \cdot -1 = -16\):
   \[
   = 25 + 16 = 41
   \]

5. **Combine the results**:
   Now we have:
   \[
   \frac{-24 + 11i}{41} = -\frac{24}{41} + \frac{11}{41}i
   \]

### Final Answer
The simplified form of the expression is:
\[
-\frac{24}{41} + \frac{11}{41}i
\]

Would you like further details, or do you have any questions?

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Here are 5 related questions to explore further:

1. How do you find the conjugate of a complex number?
2. Why does multiplying by the conjugate remove the imaginary part from the denominator?
3. What is the significance of expressing complex numbers in the form \(a + bi\)?
4. How can this simplification method be applied to other fractions involving complex numbers?
5. What happens if you multiply two complex conjugates together?

**Tip**: When simplifying complex fractions, always consider using the conjugate of the denominator to eliminate \(i\) from the denominator.

Simplify (4 + i) / (-5 - 4i) and write the answer in the form a + bi. Reduce all fractions.

Learn how to simplify the complex fraction (4 + i) / (-5 - 4i) by using the conjugate of the denominator. This solution includes step-by-step simplification to express the result in the form a + bi. Suitable for students in Grades 10-12.

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