Multiply $ \color{blue}{3} $ by $ \left( y-2\right) $ $$ \color{blue}{3} \cdot \left( y-2\right) = 3y-6 $$

Multiply $ \dfrac{4}{3y-6} $ by $ \dfrac{2-y}{3} $ to get $ \dfrac{ -4 }{ 9 } $.

Step 1: Factor numerators and denominators.

Step 2: Cancel common factors.

Step 3: Multiply numerators and denominators.

$$ \begin{aligned} \frac{4}{3y-6} \cdot \frac{2-y}{3} & \xlongequal{\text{Step 1}} \frac{ 4 }{ 3 \cdot \color{red}{ \left( y-2 \right) } } \cdot \frac{ \left( -1 \right) \cdot \color{red}{ \left( y-2 \right) } }{ 3 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 4 }{ 3 } \cdot \frac{ -1 }{ 3 } \xlongequal{\text{Step 3}} \frac{ -4 }{ 9 } \end{aligned} $$