jmc

Expanding the first given equation using the distributive property, we have $$\begin{array}{rl}25& =(m+n+p)(mn+mp+np)\\ & =m\cdot (mn+mp+np)+n\cdot (mn+mp+np)\\ & +p\cdot (mn+mp+np)\\ & =m^{2}n+m^{2}p+mnp+m{n}^2+mnp\\ & +n^{2}p+mnp+m{p}^2+n{p}^2\\ & =3mnp+m^{2}n+m^{2}p+m{n}^2+n^{2}p+m{p}^2+n{p}^2\end{array}$$ Expanding the second given equation using the distributive property, we have $$\begin{array}{rl}4& =m^2(n+p)+n^2(m+p)+p^2(m+n)\\ & =m^{2}n+m^{2}p+m{n}^2+n^{2}p+m{p}^2+n{p}^2\end{array}$$ We substitute the equation $$4=m^{2}n+m^{2}p+m{n}^2+n^{2}p+m{p}^2+n{p}^2$$ into the expanded form of the first given equation to get $$25=3mnp+4$$ or $mnp=\boxed{7}$.