Simplify $ \dfrac{6x^2+24x+24}{8x^3+16x^2} $ to $ \dfrac{3x+6}{4x^2} $.

Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{2x+4}$.

$$ \begin{aligned} \frac{6x^2+24x+24}{8x^3+16x^2} & =\frac{ \left( 3x+6 \right) \cdot \color{blue}{ \left( 2x+4 \right) }}{ 4x^2 \cdot \color{blue}{ \left( 2x+4 \right) }} = \\[1ex] &= \frac{3x+6}{4x^2} \end{aligned} $$

Divide $ \dfrac{30x^2+60x}{4x^2} $ by $ \dfrac{3x+6}{4x^2} $ to get $ 10x$.

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

Step 2: Cancel $ \color{red}{ 4x^2 } $ in first and second fraction.

Step 3: Factor numerators and denominators.

Step 4: Cancel common factors.

Step 5: Multiply numerators and denominators.

Step 6: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{30x^2+60x}{4x^2} }{ \frac{\color{blue}{3x+6}}{\color{blue}{4x^2}} } & \xlongequal{\text{Step 1}} \frac{30x^2+60x}{4x^2} \cdot \frac{\color{blue}{4x^2}}{\color{blue}{3x+6}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{30x^2+60x}{\color{red}{1}} \cdot \frac{\color{red}{1}}{3x+6} \xlongequal{\text{Step 3}} \frac{ 10x \cdot \color{blue}{ \left( 3x+6 \right) } }{ 1 } \cdot \frac{ 1 }{ 1 \cdot \color{blue}{ \left( 3x+6 \right) } } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ 10x }{ 1 } \cdot \frac{ 1 }{ 1 } \xlongequal{\text{Step 5}} \frac{ 10x \cdot 1 }{ 1 \cdot 1 } \xlongequal{\text{Step 6}} \frac{ 10x }{ 1 } =10x \end{aligned} $$