What is the minimum value of $x+y$? 
Suppose $x,y$ are positive real numbers that satisfy $$xy(x+2y)=2$$ What is the minimum value of $x+y$?


My Thoughts
I’ve attempted using arithmetic-geometric mean inequality and got:
$\frac{x+y+x+2y}{3} \geq \sqrt[3]{2}$
Therefore $2(x+y)+y \geq 3\sqrt[3]{2}$, then I got trapped.
Feels like I’m in the wrong way, I need a hint.

$xy(x+2y)=2$.
Let z=x+y,
$(z-y)y(z+y)=2$
$(z^2-y^2)y=2 $
$z^2-y^2=2/y $
$z^2=2/y+y^2$
$z=(2/y+y^2)^{0.5}$
$\frac {d}{dx} (2/y-y^2)^{0.5}=\frac{y^3-1}{y^2((y^3+2)/y)^{0.5}} $which equals $0$ at $y=1$
$z^2=2/1+1=3$, $z=3^{0.5}$, $x=3^{0.5}-1$,
$x+y=3^{0.5}$
