Multiply $ \dfrac{x+3}{4x-12} $ by $ \dfrac{x^2-4x+3}{x-2} $ to get $ \dfrac{x^2+2x-3}{4x-8} $.

Step 1: Factor numerators and denominators.

Step 2: Cancel common factors.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{x+3}{4x-12} \cdot \frac{x^2-4x+3}{x-2} & \xlongequal{\text{Step 1}} \frac{ x+3 }{ 4 \cdot \color{red}{ \left( x-3 \right) } } \cdot \frac{ \left( x-1 \right) \cdot \color{red}{ \left( x-3 \right) } }{ x-2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ x+3 }{ 4 } \cdot \frac{ x-1 }{ x-2 } \xlongequal{\text{Step 3}} \frac{ \left( x+3 \right) \cdot \left( x-1 \right) }{ 4 \cdot \left( x-2 \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ x^2-x+3x-3 }{ 4x-8 } = \frac{x^2+2x-3}{4x-8} \end{aligned} $$