${x}^{2}y\to $ max,

such that ${x}^{2}+4xy\le 1,x\ge 0$ and $y\ge 0$.

I think I need to use the KKT conditions here. I did however not yet succeed in solving it, so could someone please give me an example of how this should be done? And should I include the constraints $x\ge 0$ and $y\ge 0$ into the Lagrangian function?

such that ${x}^{2}+4xy\le 1,x\ge 0$ and $y\ge 0$.

I think I need to use the KKT conditions here. I did however not yet succeed in solving it, so could someone please give me an example of how this should be done? And should I include the constraints $x\ge 0$ and $y\ge 0$ into the Lagrangian function?