<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] Hint 
Define $U=2x+y$  in first equation 
You will get
$1+\sqrt{U+1}=4 U^2+\sqrt{3U}$ solve $U$
here you need to  solve $U$
$$(x+1) \sqrt{2 x^2-x+4}+8 x^2+4xy=4$$
$$(x+1) \sqrt{2 x^2-x+4}+4x(2x+y)=4$$
Then put U in second equation and find $x$
$$(x+1) \sqrt{2 x^2-x+4}+4xU=4$$
after finding $x$ , you can get $y$ from $U=2x+y$ 
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