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The image contains a question asking to find the real part and the imaginary part of complex number expressions.

We are given:
- \( z_1 = 2 - i \)
- \( z_2 = -2 + i \)

We need to find:
1. \( \text{Re} \left( \frac{z_1^2 z_2}{z_1} \right) \) (the real part of the given expression)
2. \( \text{Im} \left( \frac{1}{z_1 \bar{z_1}} \right) \) (the imaginary part of another expression involving the conjugate of \( z_1 \))

### Solution Outline:
#### Part (i): \( \text{Re} \left( \frac{z_1^2 z_2}{z_1} \right) \)
Simplify the expression:
\[
\frac{z_1^2 z_2}{z_1} = z_1 z_2
\]
We just need to compute \( z_1 z_2 \) and find its real part.

#### Part (ii): \( \text{Im} \left( \frac{1}{z_1 \bar{z_1}} \right) \)
Simplify the expression:
\[
z_1 \bar{z_1} = |z_1|^2
\]
so it becomes:
\[
\text{Im} \left( \frac{1}{|z_1|^2} \right)
\]
Since \( |z_1|^2 \) is a real number, its reciprocal is also real, and thus the imaginary part will be 0.

I'll now walk through both steps for these calculations.

---

### Solution:
#### Part (i):
1. First, multiply \( z_1 = 2 - i \) by \( z_2 = -2 + i \):
   \[
   z_1 z_2 = (2 - i)(-2 + i)
   \]
   Using distributive property:
   \[
   = 2(-2 + i) - i(-2 + i)
   = -4 + 2i + 2i + 1
   = -3 + 4i
   \]
   Therefore, \( z_1 z_2 = -3 + 4i \).
   
2. The real part is:
   \[
   \text{Re}(-3 + 4i) = -3
   \]

#### Part (ii):
1. The conjugate of \( z_1 = 2 - i \) is \( \bar{z_1} = 2 + i \).
   
2. Now, compute \( z_1 \bar{z_1} \):
   \[
   z_1 \bar{z_1} = (2 - i)(2 + i) = 4 - i^2 = 4 + 1 = 5
   \]
   Therefore, \( z_1 \bar{z_1} = 5 \).

3. The given expression simplifies to:
   \[
   \frac{1}{z_1 \bar{z_1}} = \frac{1}{5}
   \]
   Since \( \frac{1}{5} \) is a real number, its imaginary part is 0:
   \[
   \text{Im} \left( \frac{1}{5} \right) = 0
   \]

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### Final Answers:
1. \( \text{Re} \left( \frac{z_1^2 z_2}{z_1} \right) = -3 \)
2. \( \text{Im} \left( \frac{1}{z_1 \bar{z_1}} \right) = 0 \)

Would you like further details or have any questions? Here are some related questions:

1. What is the modulus of a complex number, and how is it used in calculations like these?
2. How do you compute the conjugate of a complex number, and why is it useful?
3. What role does the imaginary unit \(i\) play in these kinds of calculations?
4. How can you find the real and imaginary parts of a product of two complex numbers?
5. How do we simplify fractions involving complex numbers?

**Tip**: When multiplying complex numbers, always expand using distributive properties and remember that \(i^2 = -1\).

Let z1 = 2 - i, z2 = -2 + i find: (i) Re((z1^2 * z2)/z1) (ii) Im(1/(z1 * conjugate(z1))).

Explore how to find the real part of (z1^2 * z2)/z1 and the imaginary part of 1/(z1 * conjugate(z1)) with detailed step-by-step calculations. This problem delves into complex numbers and their properties, including multiplication and conjugates.

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