Simplify $ \dfrac{4x^3+24x^2-64x}{x^2+3x-10} $ to $ \dfrac{4x^2+32x}{x+5} $.

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

$$ \begin{aligned} \frac{4x^3+24x^2-64x}{x^2+3x-10} & =\frac{ \left( 4x^2+32x \right) \cdot \color{blue}{ \left( x-2 \right) }}{ \left( x+5 \right) \cdot \color{blue}{ \left( x-2 \right) }} = \\[1ex] &= \frac{4x^2+32x}{x+5} \end{aligned} $$

Simplify $ \dfrac{x^2-25}{6x^4+18x^3-240x^2} $ to $ \dfrac{x+5}{6x^3+48x^2} $.

Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{x-5}$.

$$ \begin{aligned} \frac{x^2-25}{6x^4+18x^3-240x^2} & =\frac{ \left( x+5 \right) \cdot \color{blue}{ \left( x-5 \right) }}{ \left( 6x^3+48x^2 \right) \cdot \color{blue}{ \left( x-5 \right) }} = \\[1ex] &= \frac{x+5}{6x^3+48x^2} \end{aligned} $$

Multiply $ \dfrac{4x^2+32x}{x+5} $ by $ \dfrac{x+5}{6x^3+48x^2} $ to get $ \dfrac{ 2 }{ 3x } $.

Step 1: Cancel $ \color{red}{ x+5 } $ in first and 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.

$$ \begin{aligned} \frac{4x^2+32x}{x+5} \cdot \frac{x+5}{6x^3+48x^2} & \xlongequal{\text{Step 1}} \frac{4x^2+32x}{\color{red}{1}} \cdot \frac{\color{red}{1}}{6x^3+48x^2} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 2 \cdot \color{blue}{ \left( 2x^2+16x \right) } }{ 1 } \cdot \frac{ 1 }{ 3x \cdot \color{blue}{ \left( 2x^2+16x \right) } } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2 }{ 1 } \cdot \frac{ 1 }{ 3x } \xlongequal{\text{Step 4}} \frac{ 2 \cdot 1 }{ 1 \cdot 3x } \xlongequal{\text{Step 5}} \frac{ 2 }{ 3x } \end{aligned} $$