<s>[INST] System of equations with 2 parameters I have no idea how even to start!
\begin{align*}
(u^2+v^2)(u+v)&=15uv \\
(u^4+v^4)(u^2+v^2)&=85u^2v^2
\end{align*}
 [/INST] The system 
\begin{align}
(u^2+v^2)(u+v)&=15 u v, \\
(u^4+v^4)(u^2+v^2)&=85u^2 v^2.
\end{align}
can be rewritten,
using Newton's identities:
\begin{align}
p_1 & = e_1 \\
p_2 &= e_1 p_1 - 2 e_2 \\
p_3 &= e_1 p_2 - e_2 p_1 + 3 e_3 = e_1 p_2 - e_2 p_1 \\
p_4 &=e_1 p_3-e_2 p_2 + e_3 p_1 -4 e_4 = e_1 p_3-e_2 p_2
\end{align}
as
\begin{align}
p_2 e_1&=15 e_2, \\
p_4 p_2&=85 e_2^2,
\end{align}
where
\begin{align}
p_i&=u^i+v^i, \\
e_1&=u+v,\ e_2=u v, \ e_i=0,\quad \forall i>2.
\end{align}
Now we can express $p_2,p_3$ and $p_4$ in terms of $e_1$ and $e_2$
and find out that
\begin{align}
e_2 = \frac{e_1^3}{2 e_1+15}.
\end{align}
Then it boils down to equation in terms of $e_1$:
\begin{align}
e_1^4 (3e_1+35)(e_1-6)=0,
\end{align}
hence
\begin{align}
e_1&=0,6,-\frac{35}{3}; \\
e_2&=0,8,\frac{1715}{9}.
\end{align}
Thus solutions to the original system are:
\begin{align}
(u,v)&=(0,0),(2,4),\left(\frac{-35+7i\sqrt{115}}{6},\frac{-35-7i\sqrt{115}}{6}\right),
\end{align}
and since the system is symmetric, we also need to include pairs $(v,u)$.
The illustration shows points of real solutions:

</s>