Divide $ \dfrac{x+3}{4x-12} $ by $ x-2 $ to get $ \dfrac{x+3}{4x^2-20x+24} $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{x+3}{4x-12} }{x-2} & \xlongequal{\text{Step 1}} \frac{x+3}{4x-12} \cdot \frac{\color{blue}{1}}{\color{blue}{x-2}} \xlongequal{\text{Step 2}} \frac{ \left( x+3 \right) \cdot 1 }{ \left( 4x-12 \right) \cdot \left( x-2 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x+3 }{ 4x^2-8x-12x+24 } = \frac{x+3}{4x^2-20x+24} \end{aligned} $$

Divide $ \dfrac{x+3}{4x^2-20x+24} $ by $ x^2-4x+3 $ to get $ \dfrac{x+3}{4x^4-36x^3+116x^2-156x+72} $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{x+3}{4x^2-20x+24} }{x^2-4x+3} & \xlongequal{\text{Step 1}} \frac{x+3}{4x^2-20x+24} \cdot \frac{\color{blue}{1}}{\color{blue}{x^2-4x+3}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x+3 \right) \cdot 1 }{ \left( 4x^2-20x+24 \right) \cdot \left( x^2-4x+3 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x+3 }{ 4x^4-16x^3+12x^2-20x^3+80x^2-60x+24x^2-96x+72 } = \frac{x+3}{4x^4-36x^3+116x^2-156x+72} \end{aligned} $$