jmc

We use the given information to set up a quadratic that relates the area of the field to $m$: $$\begin{array}{rl}(2m+7)(m-2)& =51\\ 2m^2+3m-14& =51\\ 2m^2+3m-65& =0\\ (2m+13)(m-5)& =0\end{array}$$The two possible solutions are $m=-\frac{13}{2}$ and $m=5$. Of these, only $m=\boxed{5}$ is valid.