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The given equation is:

\[
(z - i)^4 + (z + 2i)^4 = 0
\]
where \( z \in \mathbb{C} \), meaning \( z \) is a complex number.

### Step-by-Step Solution:

1. **Rewrite the equation:**
   The equation is the sum of two terms, both raised to the fourth power:
   \[
   (z - i)^4 + (z + 2i)^4 = 0
   \]

2. **Factor the equation:**
   To solve, recognize that this is of the form \( a^4 + b^4 = 0 \), where \( a = z - i \) and \( b = z + 2i \). 

   This can be factored using the identity for the sum of two fourth powers:
   \[
   a^4 + b^4 = (a^2 + b^2)(a^2 - b^2)
   \]

   So we have:
   \[
   (z - i)^4 + (z + 2i)^4 = 0 \implies (z - i)^2 + (z + 2i)^2 = 0
   \]
   (Note: The other factor, \( (a^2 - b^2) \), doesn't lead to real solutions here since it's always positive.)

3. **Simplify each term:**
   Now expand each term in the factorized equation:
   \[
   (z - i)^2 = z^2 - 2zi - 1
   \]
   \[
   (z + 2i)^2 = z^2 + 4zi - 4
   \]

   Now substitute into the equation:
   \[
   (z^2 - 2zi - 1) + (z^2 + 4zi - 4) = 0
   \]
   Simplify:
   \[
   2z^2 + 2zi - 5 = 0
   \]
   Divide the entire equation by 2:
   \[
   z^2 + zi - \frac{5}{2} = 0
   \]

4. **Solve the quadratic equation:**
   The quadratic equation \( z^2 + zi - \frac{5}{2} = 0 \) can be solved using the quadratic formula. For an equation of the form \( az^2 + bz + c = 0 \), the solution is:
   \[
   z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   Here, \( a = 1 \), \( b = i \), and \( c = -\frac{5}{2} \). Substituting these values:
   \[
   z = \frac{-i \pm \sqrt{i^2 - 4(1)\left(-\frac{5}{2}\right)}}{2}
   \]
   Simplifying further:
   \[
   z = \frac{-i \pm \sqrt{-1 + 10}}{2} = \frac{-i \pm \sqrt{9}}{2}
   \]
   \[
   z = \frac{-i \pm 3}{2}
   \]

5. **Final solutions:**
   Therefore, the solutions are:
   \[
   z = \frac{-i + 3}{2} \quad \text{or} \quad z = \frac{-i - 3}{2}
   \]
   Which simplifies to:
   \[
   z = \frac{3 - i}{2} \quad \text{or} \quad z = \frac{-3 - i}{2}
   \]

### Final Answer:
The two solutions to the equation are:
\[
z_1 = \frac{3 - i}{2}, \quad z_2 = \frac{-3 - i}{2}
\]

Would you like more details or have any questions on this solution?

Here are 5 related questions to further explore:
1. How would the equation change if the powers were different (e.g., squared instead of fourth power)?
2. What is the geometric interpretation of the complex solutions on the complex plane?
3. How does factoring work for complex numbers in other higher powers?
4. How does this method apply to polynomials of higher degree with complex coefficients?
5. Can this approach be used to find roots of other non-linear complex equations?

**Tip:** Always check if complex equations can be factored before expanding, as it may lead to simpler solutions.

Step-by-step solution for solving the complex equation (z - i)^4 + (z + 2i)^4 = 0. Learn how to factor fourth powers and solve quadratic equations involving complex numbers. Suitable for high school and college-level students.

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