Let $ABCD$ be a parallelogram with area 30. Points $P$ and $Q$ are the projections of $A$ and $C$, respectively, onto the line $BD$; and points $R$ and $S$ are the projections of $B$ and $D$, respectively, onto the line $AC$. See the figure, which also shows the relative locations of these points.

Suppose $PQ=8$ and $RS=10$, and let $d$ denote the length of $\overline{BD}$, the longer diagonal of $ABCD$. Then $d^2$ can be written in the form $m+n\sqrt{p}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m+n-p$?