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To simplify the expression \(\frac{2 - 7i}{4 + 5i}\), we can multiply the numerator and the denominator by the complex conjugate of the denominator. The conjugate of \(4 + 5i\) is \(4 - 5i\).

Here's the step-by-step solution:

1. **Multiply by the Conjugate:**
   \[
   \frac{2 - 7i}{4 + 5i} \cdot \frac{4 - 5i}{4 - 5i} = \frac{(2 - 7i)(4 - 5i)}{(4 + 5i)(4 - 5i)}
   \]

2. **Expand the Numerator:**
   Using the distributive property (FOIL), we get:
   \[
   (2 - 7i)(4 - 5i) = 2 \cdot 4 + 2 \cdot (-5i) - 7i \cdot 4 - 7i \cdot (-5i)
   \]
   \[
   = 8 - 10i - 28i + 35i^2
   \]
   Since \(i^2 = -1\), substitute \(35i^2 = 35(-1) = -35\):
   \[
   = 8 - 10i - 28i - 35
   \]
   Combine like terms:
   \[
   = (8 - 35) + (-10i - 28i) = -27 - 38i
   \]

3. **Expand the Denominator:**
   The denominator is a difference of squares:
   \[
   (4 + 5i)(4 - 5i) = 4^2 - (5i)^2 = 16 - 25i^2
   \]
   Substitute \(i^2 = -1\):
   \[
   = 16 - 25(-1) = 16 + 25 = 41
   \]

4. **Combine the Results:**
   Now we have:
   \[
   \frac{-27 - 38i}{41} = -\frac{27}{41} - \frac{38}{41}i
   \]

### Final Answer:
\[
\frac{2 - 7i}{4 + 5i} = -\frac{27}{41} - \frac{38}{41}i
\]

Would you like more details on any step?

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Here are five related questions that expand on this topic:

1. How do you find the conjugate of a complex number?
2. Why is it necessary to multiply by the conjugate when dividing complex numbers?
3. What are some applications of complex numbers in real-world scenarios?
4. How would this calculation change if the denominator was \(4 - 5i\) instead of \(4 + 5i\)?
5. Can we simplify complex number division using polar form instead?

**Tip:** When dividing complex numbers, always use the conjugate of the denominator to eliminate the imaginary part in the denominator.

Learn to simplify the complex fraction (2 - 7i) / (4 + 5i) by multiplying by the conjugate. This detailed solution covers complex conjugates, multiplication, and simplifying complex numbers. Suitable for students in Grades 10-12.

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