jmc

For a quadratic to have two real roots, the discriminant must be greater than 0. So we require $$\begin{array}{rl}15^2-4\cdot 8\cdot c& >0\\ \Rightarrow 225-32c& >0\\ \Rightarrow c& <\frac{225}{32}.\end{array}$$The largest integer smaller than $\frac{225}{32}$ is 7. Thus, the positive integer values of $c$ are 1, 2, 3, 4, 5, 6, and 7 and their product is $\boxed{5040}$.