1. We are asked to solve the equation $z^{5} + 1 = 0$. 2. Rewrite the equation as $z^{5} = -1$. 3. Express $-1$ in polar form: $-1 = \cos(\pi) + i\sin(\pi)$. 4. Use De Moivre's theorem to find the 5th roots of $-1$: $$z_k = \cos\left(\frac{\pi + 2k\pi}{5}\right) + i\sin\left(\frac{\pi + 2k\pi}{5}\right), \quad k=0,1,2,3,4.$$ 5. These are the 5 distinct complex roots evenly spaced on the unit circle at angles $\frac{\pi + 2k\pi}{5}$. 6. The roots can also be written as: $$z_k = e^{i\frac{\pi + 2k\pi}{5}}, \quad k=0,1,2,3,4.$$ This completes the solution for the first problem.