### Instruction: Find the product of all positive integer values of $c$ such that $3x^2+7x+c=0$ has two real roots.### Response: To find the product of all positive integer values of $c$, I need to first find all such values of $c$.
I know that a quadratic equation of the form $ax^2+bx+c=0$ has two real roots if and only if its discriminant, $b^2-4ac$, is positive.
So, in this case, I need to find all positive integers $c$ such that $7^2-4\cdot 3\cdot c>0$.
Simplifying, I get $49-12c>0$, or $c<\frac{49}{12}$.
Since $c$ has to be a positive integer, the only possible values of $c$ are $1, 2, 3$, and $4$.
To find the product of these values, I just need to multiply them together.