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

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Factor numerators and denominators.

Step 3: Cancel common factors.

Step 4: Multiply numerators and denominators.

Step 5: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{x+3}{4x-12} }{ \frac{\color{blue}{x-2}}{\color{blue}{x^2-4x+3}} } & \xlongequal{\text{Step 1}} \frac{x+3}{4x-12} \cdot \frac{\color{blue}{x^2-4x+3}}{\color{blue}{x-2}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ x+3 }{ 4 \cdot \color{red}{ \left( x-3 \right) } } \cdot \frac{ \left( x-1 \right) \cdot \color{red}{ \left( x-3 \right) } }{ x-2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x+3 }{ 4 } \cdot \frac{ x-1 }{ x-2 } \xlongequal{\text{Step 4}} \frac{ \left( x+3 \right) \cdot \left( x-1 \right) }{ 4 \cdot \left( x-2 \right) } = \\[1ex] & \xlongequal{\text{Step 5}} \frac{ x^2-x+3x-3 }{ 4x-8 } = \frac{x^2+2x-3}{4x-8} \end{aligned} $$