Elegance is the key!

$mn+3m-8n-24=59-24 \implies m(n+3)-8(n+3)=35=7 \times 5 \implies (m-8)(n+3)=35=7 \times 5$

$\begin{cases} m-8=7 \implies m=15\\ n+3=5 \implies n=2 \\ \end{cases}$

$\begin{cases} m-8=5 \implies m=13\\ n+3=7 \implies n=4 \\ \end{cases}$

$\begin{cases} m-8=1 \implies m=9\\ n+3=35 \implies n=32 \\ \end{cases}$

$\begin{cases} m-8=35 \implies m=43\\ n+3=1 \implies n=-2 \\ \end{cases}$

only the last case gives a negative integer for $n$ , which is not acceptable.