Let $\alpha$ be the shared root. Then $$ \left\{ \begin{array}{ll} \alpha^2 - \alpha m + 5 + m = 0 \\ \alpha^2 - (7m+1)\alpha + 13m + 7 = 0 \end{array} \right. $$ Subtracting the second equation from the first one: $(6m+1)\alpha = 2(6m+1)$. When $(6m+1) = 0 \implies m=-\frac{1}{6}$. However in this case, the first given equation has no solution If $(6m+1)\ne 0$, then $\alpha=2\implies m=9$. By Vieta's theorem, we find the product of the four roots are $$(5+m)(13m+7)=\boxed{1736}$$