Solving a complex cubic equation I am trying to solve the following equation:
$$
z^3 + z +1=0
$$
Attempt: I tried to factor out this equation to get a polynomial term, but none of the roots of the equation is trivial.

Substitute $x=\frac2{\sqrt3}\sinh t$ to rewrite the equation  $x^3+x+1=0 $ as
$$4\sinh^3t+3\sinh t + \frac{3\sqrt3}2=0$$
Comparing with the identity $4\sinh^3t+3\sinh t = \sinh3t$
results in $\sinh 3t =- \frac{3\sqrt3}2$, or
$t = -\frac13\sinh^{-1} \frac{3\sqrt3}2$. Thus, one real root of the cubic equation is
$$x_0= -\frac2{\sqrt3}\sinh \left( \frac13\sinh^{-1} \frac{3\sqrt3}2\right) \approx -0.6823
$$
Then, factorize the cubic equation as
$$x^3+x+1=(x-x_0)(x^2+x_0 x-\frac1{x_0})$$
and the quadratic factor gives a pair of complex roots
$$x_{1,2}=-\frac{x_0}2 \pm \frac i2 \sqrt{1-\frac3{x_0}}$$
