Multiply. Write your answer in simplest form. \[ \frac{3 y-2}{3 y^{2}-2 y} \cdot \frac{8 y+16}{y+2} \]

To simplify the expression, we start by factoring where possible: 1. The first fraction is \(\frac{3y-2}{3y^2 - 2y}\). We can factor the denominator: \[ 3y^2 - 2y = y(3y - 2). \] Thus, the first fraction can be rewritten as: \[ \frac{3y - 2}{y(3y - 2)}. \] 2. Now consider the second fraction \(\frac{8y+16}{y+2}\). The numerator can be factored: \[ 8y + 16 = 8(y + 2). \] Therefore, the second fraction becomes: \[ \frac{8(y + 2)}{y + 2}. \] Now we can rewrite the entire expression: \[ \frac{3y - 2}{y(3y - 2)} \cdot \frac{8(y + 2)}{y + 2}. \] Next, we can cancel out terms: - The \((3y - 2)\) terms cancel out. - The \((y + 2)\) in the numerator of the second fraction and the denominator of the second fraction cancel out. Now we have: \[ \frac{8}{y} \] Thus, the final simplified form of the given expression is: \[ \frac{8}{y}. \]