\[\begin{array}{l}
\frac{1}{{{z^2}}} + \frac{1}{z} + 1 + z + {z^2} = 0\\
z + \frac{1}{z} = y\\
{y^2} = {z^2} + 2z\frac{1}{z} + \frac{1}{{{z^2}}} = {z^2} + 2 + \frac{1}{{{z^2}}}\\
{z^2} + \frac{1}{{{z^2}}} = {y^2} - 2\\
{z^2} + \frac{1}{{{z^2}}} + z + \frac{1}{z} + 1 = 0\\
{y^2} - 2 + y + 1 = 0\\
{y^2} + y - 1 = 0\\
napomena\\
{\left( {z + \frac{1}{z}} \right)^2} \ne {z^2} + \frac{1}{{{z^2}}}
\end{array}\]