To find all the cube roots of $-1 + \sqrt{5}i$, let's proceed by expressing this complex number in polar form and then using De Moivre's theorem.

1. Convert $-1 + \sqrt{5}i$ to polar form:

   For a complex number $z = x + yi$, we can express it in polar form as: $z = r \left( \cos \theta + i \sin \theta \right)$ where: $r = \sqrt{x^2 + y^2}$ and $\theta = \arctan \left( \frac{y}{x} \right).$

   For $z = -1 + \sqrt{5}i$:

   • $x = -1$
   • $y = \sqrt{5}$

   So, $r = \sqrt{(-1)^2 + (\sqrt{5})^2} = \sqrt{1 + 5} = \sqrt{6}.$

   Now, calculate $\theta$: $\theta = \arctan \left( \frac{\sqrt{5}}{-1} \right).$ Since $-1 + \sqrt{5}i$ lies in the second quadrant, we adjust the angle accordingly: $\theta = \pi + \arctan \left( -\frac{\sqrt{5}}{1} \right).$ Let’s denote this as $\theta_0$ for simplicity.

2. Apply the formula for cube roots:

   The cube roots of $z = r \left( \cos \theta + i \sin \theta \right)$ are given by: $\sqrt[3]{r} \left( \cos \frac{\theta + 2k\pi}{3} + i \sin \frac{\theta + 2k\pi}{3} \right)$ for $k = 0, 1, 2$.

   • Here, $\sqrt[3]{r} = \sqrt[3]{\sqrt{6}} = \sqrt[6]{6}$.
   • For $k = 0, 1, 2$, we get different values of the angle.

3. Calculate the cube roots:

   For each $k$: $z_k = \sqrt[6]{6} \left( \cos \frac{\theta_0 + 2k\pi}{3} + i \sin \frac{\theta_0 + 2k\pi}{3} \right).$

   Now we can substitute the values for each $k$ to obtain the three distinct cube roots of $-1 + \sqrt{5}i$.

Would you like a more detailed calculation for each cube root value?

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Here are some questions to explore further:

1. How does converting a complex number to polar form help in finding its roots?
2. What are the steps to calculate the magnitude and angle for a complex number?
3. How does De Moivre's theorem simplify the process of finding roots of complex numbers?
4. Can the cube roots of a complex number have both real and imaginary components?
5. How does the choice of $k$ values affect the distinctiveness of roots in complex numbers?

Tip: When finding roots of complex numbers, always ensure the angle is correctly placed in the appropriate quadrant for accurate polar representation.