What is the product? \( \frac{2 y}{y-3} \cdot \frac{4 y-12}{2 y+6} \) \( \frac{2}{3} \) \( \frac{4 y}{9} \) \( \frac{4 y}{y-3} \)

To find the product of the given expressions, let's simplify it step by step. First, notice that \( 4y - 12 \) can be factored as \( 4(y - 3) \). The expression now looks like this: \[ \frac{2y}{y - 3} \cdot \frac{4(y - 3)}{2y + 6} \] Next, simplify \( 2y + 6 \) as \( 2(y + 3) \). Substituting this back into our expression gives: \[ \frac{2y}{y - 3} \cdot \frac{4(y - 3)}{2(y + 3)} \] Now, we can cancel the \( (y - 3) \) from the numerator and denominator: \[ \frac{2y \cdot 4}{2(y + 3)} \] This simplifies to: \[ \frac{8y}{2(y + 3)} = \frac{4y}{y + 3} \] Since this expression isn't in the answer choices provided, we can make one last check on the calculation and see that \( \frac{4y}{y + 3} \) would lead us to none of the answers given, but we will keep simplifying the possible product up to: The answer choices included \( \frac{4y}{9} \) and \( \frac{2}{3} \). The correct final product based on the given expressions does simplify directly to \( \frac{4y}{y - 3} \) due to the initial cancelation hits of shared numbers, resulting in our engaged final evaluation: The correct product is: \(\frac{4y}{y + 3}\) Thus, this suggests that you would also want to check if the expected further inputs have been provided as actual expected outputs based on rational simplifications. It seems none of above lead to final \( \frac{4y}{y-3} \).