Find the product of all positive integer values of c such that \(8x^2+15x+c=0 \) has two real roots.
For a quadratic to have two real roots, the discriminant must be greater than 0. So we require
\(\begin{align*}15^2-4 \cdot 8 \cdot c &> 0 \\ \Rightarrow \quad 225-32c &> 0 \\ \Rightarrow \quad c&< \frac{225}{32}.\end{align*}\)
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 \(\boxed{5040}\).
