<s>[INST] Find the three roots of the equation $\left(z+1\right)^3=-1$ 
Give your answers in the form $x+ iy$, where $x\in \mathbb{R}\:and\:y\in \mathbb{R}$.

I suspect I'll have to use de Moivre's theorem to solve this, but I don't know how to factor in the $+1$, because increasing the real part of a complex number by one doesn't necessarily increase its modulus by one.
I know that one of the roots will be $-2$, by inspection, but that's about it.
 [/INST] Consider instead 
$$c^3 = -1$$
where $c = z + 1$. Denoting in geometric form $c = re^{i\theta}$ and $-1 = e^{i \pi}$, we get
$$r^3 e^{i3\theta} = e^{i\pi + 2k\pi}, \qquad k \in \lbrace 0,1,2 \rbrace$$ 
We get $r = 1$ and three different angles 
\begin{align}
\theta_1 &= \frac{\pi}{3} =  \\
\theta_2 &= \frac{\pi}{3} + \frac{2\pi}{3} = \pi\\
\theta_3 &= \frac{\pi}{3} + \frac{4\pi}{3} = \frac{5\pi}{3}\\
\end{align}
So we get $c_1 = e^{i\frac{\pi}{3}}$, $c_2 = e^{i\pi}$ and $c_3 = e^{i\frac{5\pi}{3}}$. The corresponding roots are
\begin{align}
z_1 &= -1 + e^{i\frac{\pi}{3}} = (-1 + \cos(\frac{\pi}{3})) + i\sin(\frac{\pi}{3})  \\
z_2 &= -1 + e^{i\pi} = (-1 + \cos(\pi)) + i\sin(\pi)\\
z_3 &= -1 + e^{i\frac{5\pi}{3}}= (-1 + \cos(\frac{5\pi}{3})) + i\sin(\frac{5\pi}{3})\\
\end{align}
</s>