Multiply $ \dfrac{2x+3}{x-5} $ by $ \dfrac{x-5}{4x+6} $ to get $ \dfrac{ 1 }{ 2 } $.

Step 1: Cancel $ \color{red}{ x-5 } $ in first and second fraction.

Step 2: Factor numerators and denominators.

Step 3: Cancel common factors.

Step 4: Multiply numerators and denominators.

$$ \begin{aligned} \frac{2x+3}{x-5} \cdot \frac{x-5}{4x+6} & \xlongequal{\text{Step 1}} \frac{2x+3}{\color{red}{1}} \cdot \frac{\color{red}{1}}{4x+6} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 1 \cdot \color{blue}{ \left( 2x+3 \right) } }{ 1 } \cdot \frac{ 1 }{ 2 \cdot \color{blue}{ \left( 2x+3 \right) } } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 1 }{ 1 } \cdot \frac{ 1 }{ 2 } \xlongequal{\text{Step 4}} \frac{ 1 }{ 2 } \end{aligned} $$