Disclaimer: The solutions we've shared are just one exciting approach, and there are surely many other wonderful methods out there. We’d love to hear your alternative solutions in the community thread below, so let's keep the creativity flowing!

$$ (x+2y)^2+2x+5y+9-(x+2y+1)^2 = y+8$$

If $x,y,z$ are positive, then $(x+2y)^2+2x+5y+9 > (x+2y+1)^2$.

$$(x+2y+2)^2-{(x+2y)^2+2x+5y+9} = -5 + 2x + 3y \geq  0$$

$$\implies (x+2y+2)^2 \geq {(x+2y)^2+2x+5y+9} > (x+2y+1)^2$$

So, $(x+2y+2)^2=(x+2y)^2+2x+5y+9$ or $2x+3y-5=0$. To make this condition true, $x=1,y=1$. By substituting $x=1,y=1$, we obtain $z=4$ and we are done.