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The problem is asking for the reciprocal (Kehrwert) of \( z_1 \) and the quotient \( \frac{z_1}{z_2} \), where the complex numbers are given as:

\[
z_1 = -3 - j
\]
\[
z_2 = -4 + 5j
\]

### Step 1: Calculate the Reciprocal of \( z_1 \)

To find the reciprocal of \( z_1 \), we use the formula:
\[
\frac{1}{z_1} = \frac{\overline{z_1}}{|z_1|^2}
\]
where \( \overline{z_1} \) is the complex conjugate of \( z_1 \) and \( |z_1|^2 \) is the modulus squared of \( z_1 \).

1. **Complex conjugate of \( z_1 \):** \( \overline{z_1} = -3 + j \)
2. **Modulus squared of \( z_1 \):**
   \[
   |z_1|^2 = (-3)^2 + (-1)^2 = 9 + 1 = 10
   \]
3. **Reciprocal of \( z_1 \):**
   \[
   \frac{1}{z_1} = \frac{-3 + j}{10} = -0.3 + 0.1j
   \]

So, the reciprocal \( \frac{1}{z_1} = -0.3 + 0.1j \).

### Step 2: Calculate the Quotient \( \frac{z_1}{z_2} \)

To find \( \frac{z_1}{z_2} \), we use the formula:
\[
\frac{z_1}{z_2} = \frac{z_1 \cdot \overline{z_2}}{|z_2|^2}
\]
where \( \overline{z_2} \) is the complex conjugate of \( z_2 \) and \( |z_2|^2 \) is the modulus squared of \( z_2 \).

1. **Complex conjugate of \( z_2 \):** \( \overline{z_2} = -4 - 5j \)
2. **Modulus squared of \( z_2 \):**
   \[
   |z_2|^2 = (-4)^2 + (5)^2 = 16 + 25 = 41
   \]
3. **Calculate \( z_1 \cdot \overline{z_2} \):**
   \[
   z_1 \cdot \overline{z_2} = (-3 - j)(-4 - 5j)
   \]
   Expanding this product:
   \[
   = (-3)(-4) + (-3)(-5j) + (-j)(-4) + (-j)(-5j)
   \]
   \[
   = 12 + 15j + 4j + 5j^2
   \]
   Since \( j^2 = -1 \):
   \[
   = 12 + 15j + 4j - 5 = 7 + 19j
   \]
4. **Quotient \( \frac{z_1}{z_2} \):**
   \[
   \frac{z_1}{z_2} = \frac{7 + 19j}{41} = \frac{7}{41} + \frac{19}{41}j \approx 0.17 + 0.46j
   \]

### Final Answers
- Reciprocal \( \frac{1}{z_1} = -0.3 + 0.1j \)
- Quotient \( \frac{z_1}{z_2} \approx 0.17 + 0.46j \)

Would you like further details on any step?

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**Related Questions:**

1. How do you find the modulus of a complex number?
2. What is the significance of the complex conjugate in division of complex numbers?
3. How would you express the reciprocal of a complex number in polar form?
4. Can the reciprocal of a complex number be real?
5. How does complex division relate to polar coordinates?

**Tip:** When dividing complex numbers, multiplying by the conjugate of the denominator simplifies the division process.

(c) Berechnen Sie den Kehrwert \( \frac{1}{z_1} \) und den Quotienten \( \frac{z_1}{z_2} \) zu \( z_1 = -3 - j \) und \( z_2 = -4 + 5j \).

Learn how to calculate the reciprocal and quotient of the complex numbers \( z_1 = -3 - j \) and \( z_2 = -4 + 5j \). This solution includes key steps, such as finding the complex conjugate and modulus, and applying formulas for division and reciprocals in complex numbers. Suitable for students in Grades 11-12.

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