Find the product of all positive integer values of $c$ such that the quadratic equation $3x^2+7x+c=15x-10$ has two real roots.
We want the quadratic formula to yield two real roots, so the discriminant must be greater than $0.$ This gives us
\[b^2 - 4ac = 7^2 - 4 \cdot 3(c - 15) > 0.\]Then $49 - 12c + 180 > 0,$ so $c < \frac{229}{12}.$
Also, $c = 15x - 10 - 3x^2 - 7x = -3x^2 + 8x - 10,$ so the quadratic $-3x^2 + 8x + (10 - c) = 0$ must have two real roots.
A quadratic has two real roots if and only if its discriminant is nonnegative, so
\[8^2 - 4(-3)(10 - c) \ge 0.\]Then $64 - 12(10 - c) \ge 0,$ so $c \ge 8.$
The possible values of $c$ are then 8, 9, 10, $\dots,$ 18. The product of all positive integer values of $c$ from 8 to $\left\lfloor \frac{229}{12} \right\rfloor = 19$ is $\boxed{901,536}.$
To find the product of all positive integer values of $c$, we need to solve the equation $3x^2 + 7x + c = 15x - 10$ and determine the values of $c$ that result in two real roots.
First, let's rewrite the equation in standard form by moving all terms to one side:
$3x^2 + 7x + c = 15x - 10$
Rearranging the terms, we have:
$3x^2 - 8x + c + 10 = 0$
For a quadratic equation to have two real roots, the discriminant must be greater than zero. The discriminant, denoted by $\Delta$, is given by the formula $\Delta = b^2 - 4ac$.
In our equation, $a = 3$, $b = -8$, and $c = c + 10$. We want to find the product of all positive integer values of $c$, so let's proceed with the calculation:
$\Delta = (-8)^2 - 4(3)(c + 10)$
Simplifying further:
$\Delta = 64 - 12(c + 10)$
$\Delta = 64 - 12c - 120$
$\Delta = 192 - 12c$
Now, we set the discriminant greater than zero:
$\Delta > 0$
$192 - 12c > 0$
$192 > 12c$
$c < 16$
Since we need to find positive integer values of $c$, we can now list the values from $1$ to $15$, inclusive, and calculate their product:
$1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10 \cdot 11 \cdot 12 \cdot 13 \cdot 14 \cdot 15$
Using a calculator or computing the product manually, we find that the product of all positive integer values of $c$ is $1,307,674,368,000$ (approximately).
Therefore, the product of all positive integer values of $c$ is $1,307,674,368,000$.