Multiply. State the product in simplest form. \( \frac{45 y}{y^{2}-4} \cdot \frac{y^{2}+2 y}{18 y^{2}} \) \( \frac{5}{2 y-4} \) \( \frac{-5}{4} \) \( \frac{10}{y-2} \) \( \frac{5 y}{2 y^{2}-4 y} \)

To multiply the fractions, first factor where possible: 1. The denominator \(y^2 - 4\) can be factored as \((y - 2)(y + 2)\). 2. The numerator \(y^2 + 2y\) can be factored out as \(y(y + 2)\). 3. The denominator \(18y^2\) remains as is. Now the expression becomes: \[ \frac{45y}{(y - 2)(y + 2)} \cdot \frac{y(y + 2)}{18y^2} \] Now let's combine the fractions: \[ \frac{45y \cdot y(y + 2)}{(y - 2)(y + 2) \cdot 18y^2} \] This simplifies to: \[ \frac{45y^2(y + 2)}{18y^2(y - 2)(y + 2)} \] Since \(y + 2\) in the numerator and denominator cancels out, we have: \[ \frac{45y^2}{18y^2(y - 2)} \] Now, \(y^2\) cancels as well, leaving us with: \[ \frac{45}{18(y - 2)} \] Which can be simplified further: \[ \frac{5}{2(y - 2)} \] Thus, the simplest form of the product is: \[ \frac{5}{2(y - 2)} \] So, the correct option is \( \frac{5}{2 y - 4} \).