Divide $ \dfrac{x^2-5x+6}{6x^2} $ by $ x^2+x-6 $ to get $ \dfrac{ x-3 }{ 6x^3+18x^2 } $.

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.

$$ \begin{aligned} \frac{ \frac{x^2-5x+6}{6x^2} }{x^2+x-6} & \xlongequal{\text{Step 1}} \frac{x^2-5x+6}{6x^2} \cdot \frac{\color{blue}{1}}{\color{blue}{x^2+x-6}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x-3 \right) \cdot \color{blue}{ \left( x-2 \right) } }{ 6x^2 } \cdot \frac{ 1 }{ \left( x+3 \right) \cdot \color{blue}{ \left( x-2 \right) } } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x-3 }{ 6x^2 } \cdot \frac{ 1 }{ x+3 } \xlongequal{\text{Step 4}} \frac{ \left( x-3 \right) \cdot 1 }{ 6x^2 \cdot \left( x+3 \right) } = \\[1ex] & \xlongequal{\text{Step 5}} \frac{ x-3 }{ 6x^3+18x^2 } \end{aligned} $$

Divide $ \dfrac{x-3}{6x^3+18x^2} $ by $ 2x $ to get $ \dfrac{ x-3 }{ 12x^4+36x^3 } $.

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.

$$ \begin{aligned} \frac{ \frac{x-3}{6x^3+18x^2} }{2x} & \xlongequal{\text{Step 1}} \frac{x-3}{6x^3+18x^2} \cdot \frac{\color{blue}{1}}{\color{blue}{2x}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x-3 \right) \cdot 1 }{ \left( 6x^3+18x^2 \right) \cdot 2x } \xlongequal{\text{Step 3}} \frac{ x-3 }{ 12x^4+36x^3 } \end{aligned} $$