Extreme Quadrilateral Areas

Geometry Level 3

We know by Heron's formula that if we know all the side lengths of a triangle, then we can determine the area of the triangle. However, this is not the case for a quadrilateral.

Find the maximum possible area of a quadrilateral with side lengths 3, 4, 5, and 6.

If this area can be expressed as $P \sqrt Q$ for integers $P$ and $Q$ , with $Q$ square-free, find $P+Q.$