problem: Find the product of all positive integer values of $c$ such that $3x^2+7x+c=0$ has two real roots.
solution: For a quadratic to have two real roots, the discriminant must be greater than 0. So we require \begin{align*}7^2-4 \cdot 3 \cdot c &> 0 \quad \Rightarrow \\ 49-12c &>0\quad \Rightarrow \\ c&<\frac{49}{12}.\end{align*}The largest integer smaller than $\frac{49}{12}$ is 4. Thus, the positive integer values of $c$ are 1, 2, 3, and 4, and their product is $\boxed{24}$.