OwO, it's a typo: it's suppossed to be width, okay?
If you use the AMGM theorem, you'll know that any rectangle achieves the maximum possible area when both of their measurements are equal, making it a square (a square is technically a rectangle too)
So we have p=q
Therefore, the equation will be q²+q=13+q²
<=>q=13(cm)
<=>p=q=13(cm)
<=>S=pq=13²=169(cm²)

(P.S: don't do this here, cuz you'll get to a dead end, oof)
\(pq+q=13+q^2\)
\(\Leftrightarrow S=q^2-q+13\)
 

OMG OMG THANK YOU SO VERY MUCH!!!!!!!!!!!!!!!!!!!!! :DDDDDDDDDDDD