<s>[INST] Simplify the polynomial $xy(x+y)+(x+y)+(x+y)^2=13xy$ to the form $y^2=4x^3-g_{2}x-g_{3}$ 
I try to simplify a polynomial to the form: $y^2=4x^3-g_{2}x-g_{3}$, which is the elliptic curves. And the polynomial is $xy(x+y)+(x+y)+(x+y)^2=13xy$. 

I try to let the $u=x+y$ and $v=x-y$, then I get the $u^3-uv^2+4u-9u^2+13v^2=0$. But how to get ahead?
 [/INST] Are you certain that you have stated the problem correctly? As currently stated
$$ xy(x+y)+(x+y)+(x+y)^2=13xy $$
can be rewritten
\begin{eqnarray}
x^2y+xy^2+x+y+x^2+2xy+y^2&=&13xy\\
y^2(x+1)+y(x^2-11x+1)+x&=&0\\
y(x^2-11x+1)&=&-[y^2(x+1)+x]\\
y^2(x^2-11x+1)^2&=&[y^2(x+1)+x]^2
\end{eqnarray}
So if there exist constants $g_3,\,g_2$ such that
$$y^2=4x^3-g_{2}x-g_{3}$$
then
$$ (4x^3-g_{2}x-g_{3})(x^2-11x+1)^2=[(4x^3-g_{2}x-g_{3})(x+1)+x]^2 $$
This will resolve in an eighth degree polynomial equation of the form
$$ 16x^8-4x^7+P_6(x,g_2,g_3)=0 $$
Where $P_6$ is a degree six polynomial whose cofficients are function of $g_3,\,g_2$. But the coefficients of $x^8$ and $x^7$ are independent of the values of $g_3,\,g_2$.
So no constant values of $g_3,\,g_2$ can result in a solution for $x\ne0$.
Thus there cannot be values of $g_3,\,g_2$ for which $y^2=4x^3-g_{2}x-g_{3}$.
</s>