<s>[INST] How to solve this system of equations? How to solve this system of equations?
$$\begin{cases}
1+\sqrt{2 x+y+1}=4 (2
   x+y)^2+\sqrt{6 x+3 y},\\
(x+1) \sqrt{2 x^2-x+4}+8 x^2+4
   x y=4.
\end{cases}$$
 [/INST] You have two equations with two variables.
$$1+\sqrt{2x+y+1}-4(2x+y)^2-\sqrt{6x+3y}=0$$
$$(x+1)\sqrt{2x^2-x+4}+8x^2+4xy-4=0$$
Solve it with any root finding algortihm.
If you are looking for real solutions
$$x=0.5\qquad y=-0.5$$
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