How to solve the equation $- x^4 - 2 x^3 + 14 x^2 - 2 x - 1=0$ This equation:
$- x^4 - 2 x^3 + 14 x^2 - 2 x - 1 = 0$
has 4 real solutions (I saw it on GeoGebra), but I can't calculate them analytically.
I know the solving formula of the fourth degree equations (https://it.wikipedia.org/wiki/Equazione_di_quarto_grado) :
${\displaystyle x_{1,2}=-{\frac {b}{4a}}-Q\pm {\frac {1}{2}}{\sqrt {-4Q^{2}-2p+{\frac {S}{Q}}}}}$
${\displaystyle x_{3,4}=-{\frac {b}{4a}}+Q\pm {\frac {1}{2}}{\sqrt {-4Q^{2}-2p-{\frac {S}{Q}}}}}$
with
$p={\frac{8ac-3b^{2}}{8a^{2}}}\qquad \qquad {\color {white}.}$
${\displaystyle S={\frac {8a^{2}d-4abc+b^{3}}{8a^{3}}}}$
and
${\displaystyle Q={\frac {1}{2}}{\sqrt {-{\frac {2}{3}}\ p+{\frac {1}{3a}}\left(\Delta _{0}+{\frac {q}{\Delta _{0}}}\right)}}}$
${\displaystyle \Delta _{0}={\sqrt[{3}]{\frac {s+{\sqrt {s^{2}-4q^{3}}}}{2}}}}$
and
${\displaystyle q=12ae-3bd+c^{2}}$
${\displaystyle s=27ad^{2}-72ace+27b^{2}e-9bcd+2c^{3}}$

I calculate "q" and comes "196"
I calculate "s" and comes "3760"
But when I try to calculate Δ₀, in the inner root $ \sqrt{s ^ {2} -4q ^ {3}} $ I get a negative value, precisely "- 15980544"

First note that
\begin{eqnarray*}
-x^{4}-2x^{3}+14x^{2}-2x-1=0 &\implies& x^{4}+2x^{3}-14x^{2}+2x+1=0\\
&\implies& \left(x+\frac{1}{x}\right)^{2}+2\left(x+\frac{1}{x}\right)-16=0\\
&\implies& u^{2}+2u-16=0.
\end{eqnarray*}
Then we have a quadratic equation with solutions $$u=-1\pm\sqrt{17}.$$
Therefore all the solutions are given by
$$x = \frac{1}{2}\left(-1 \pm \sqrt{17} \pm \sqrt{14 - 2 \sqrt{17}}\right).$$
