339. Solve the equation:

$$
13 x^{2}=x^{4}+2 x^{3}+2 x+1
$$

(solved elementarily).
339. Adding $3 x^{2}$ to both sides:

$$
\begin{gathered}
16 x^{2}=x^{4}+2 x^{3}+3 x^{2}+2 x+1 \\
16 x^{2}=x^{4}+2 x^{3}+2 x^{2}+x^{2}+2 x+1=\left(x^{2}+x+1\right)^{2} \\
4 x=x^{2}+x+1 \\
x^{2}-3 x+1=0 ; x=\frac{3}{2} \pm \frac{\sqrt{5}}{2}
\end{gathered}
$$

Cardano only has these two roots. The others are found from the equation:

$$
\begin{gathered}
-4 x=x^{2}+x+1 ; x^{2}+5 x+1=0 \\
x=\frac{-5 \pm \sqrt{21}}{2}
\end{gathered}
$$