The rectangle $PQRS$ is formed under the curve

The point $P$ lies on the curve.

The point $Q$ has $x$ coordinate $4$ and lies on the same horizontal line as $P.$

The points $S$ and $R$ lie on the $x$ axis directly beneath $P$ and $Q$ respectively.

Find the largest possible area for the rectangle.

Here is a diagram:

Try moving $P$ to get a sense for the problem.

If you let $P=(p,\sqrt{p})$ then you can find the area $A$ of the triangle in terms of $p.$

The solution is of the form

for some $m,n\in \mathbb{N}$. Find $m^2+n^2$.