Q:

Finding the maximum value given two system of equations

I was given $(x,y)$ that satisfies both of this equation:
$4|xy| - y^2 - 2 = 0\\(2x+y)^2 + 4x^2 = 2$
And was asked to find the maximum value of $4x + y$.
Solving for $y^2$, I get this equation:
$8x^2 + 4|xy| + 4xy - 4 = 0$
I assumed that if I were to find the maximum value of $4x + y$, then both $x$ and $y$ must be positive, so $xy \ge 0$ means that $|xy|$ is equal to $xy$. So, I have
$8x^2 + 8xy - 4 = 0$
Solving for $y$, I have $y = \frac{4-8x^2}{8x}$ which makes $4x+y = f(x) = \frac{24x^2 + 4}{8x}$. Finding the extreme point using $f'(x) = 0$, I get $x = \frac{1}{\sqrt{6}}$, $y = \frac{\sqrt{6}}{3}$, and $f(\frac{1}{\sqrt{6}}) = \sqrt{6}$, which is the wrong solution (and this being the minimum point, too, while not satisfying the first equation at the same time ...)
The choices were:
$1, \sqrt{2}, 2, \sqrt{3}, 4$. Some help would be helpful!

A:

From the two equations you get
$$4|xy|-y^2=(2x+y)^2+4x^2 \implies 2|xy|=4x^2+y^2+2xy.$$
If $xy \geq 0$, then we get $$2xy=4x^2+y^2+2xy \implies 4x^2+y^2=0 \implies x=y=0.$$
In which case the expression $\boxed{\color{blue}{4x+y=0}}$.
If $xy < 0$, then we get $$-2xy=4x^2+y^2+2xy \implies 4x^2+y^2+4xy=0 \implies (2x+y)^2=0.$$
Thus $2x+y=0$. Now the expression $\color{blue}{4x+y}=2x+(2x+y)=\boxed{\color{red}{2x}}$.
But with $2x+y=0$, we also get $y=-2x$ and plugging that in one of the given equations we get  $x=\pm\frac{1}{\sqrt{2}}$.
So the max. value of the expression $\color{blue}{4x+y}=\color{red}{2x}=\sqrt{2}$.

