jmc

The expansion of $(mx+n{)}^2$ is $m^2{x}^2+2mnx+n^2$. If $$16x^2+36x+56=m^2{x}^2+2mnx+n^2,$$we can equate the coefficients of $x^2$ to get $16=m^2$, and thus $m=4$ (we ignore the solution $-4$ because we are given that $m$ is positive).

We can also equate the coefficients of $x$ above to get $36=2mn=2(4)n=8n$, and thus $n=\frac{36}{8}=\frac{9}{2}$.

So, $mn=4\cdot \frac{9}{2}=\boxed{18}$.