Multiply $ \dfrac{8x+6}{x^2-7x+6} $ by $ x $ to get $ \dfrac{ 8x^2+6x }{ x^2-7x+6 } $.

Step 1: Write $ x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{8x+6}{x^2-7x+6} \cdot x & \xlongequal{\text{Step 1}} \frac{8x+6}{x^2-7x+6} \cdot \frac{x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( 8x+6 \right) \cdot x }{ \left( x^2-7x+6 \right) \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 8x^2+6x }{ x^2-7x+6 } \end{aligned} $$

Divide $ \dfrac{x^2-x-12}{4x^2+11x+6} $ by $ 2x+6 $ to get $ \dfrac{ x-4 }{ 8x^2+22x+12 } $.

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.

$$ \begin{aligned} \frac{ \frac{x^2-x-12}{4x^2+11x+6} }{2x+6} & \xlongequal{\text{Step 1}} \frac{x^2-x-12}{4x^2+11x+6} \cdot \frac{\color{blue}{1}}{\color{blue}{2x+6}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x-4 \right) \cdot \color{blue}{ \left( x+3 \right) } }{ 4x^2+11x+6 } \cdot \frac{ 1 }{ 2 \cdot \color{blue}{ \left( x+3 \right) } } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x-4 }{ 4x^2+11x+6 } \cdot \frac{ 1 }{ 2 } \xlongequal{\text{Step 4}} \frac{ \left( x-4 \right) \cdot 1 }{ \left( 4x^2+11x+6 \right) \cdot 2 } = \\[1ex] & \xlongequal{\text{Step 5}} \frac{ x-4 }{ 8x^2+22x+12 } \end{aligned} $$

Multiply $ \dfrac{8x+6}{x^2-7x+6} $ by $ x $ to get $ \dfrac{ 8x^2+6x }{ x^2-7x+6 } $.

Step 1: Write $ x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{8x+6}{x^2-7x+6} \cdot x & \xlongequal{\text{Step 1}} \frac{8x+6}{x^2-7x+6} \cdot \frac{x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( 8x+6 \right) \cdot x }{ \left( x^2-7x+6 \right) \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 8x^2+6x }{ x^2-7x+6 } \end{aligned} $$

Divide $ \dfrac{x-4}{8x^2+22x+12} $ by $ x^2-7x+6 $ to get $ \dfrac{x-4}{8x^4-34x^3-94x^2+48x+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.

$$ \begin{aligned} \frac{ \frac{x-4}{8x^2+22x+12} }{x^2-7x+6} & \xlongequal{\text{Step 1}} \frac{x-4}{8x^2+22x+12} \cdot \frac{\color{blue}{1}}{\color{blue}{x^2-7x+6}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x-4 \right) \cdot 1 }{ \left( 8x^2+22x+12 \right) \cdot \left( x^2-7x+6 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x-4 }{ 8x^4-56x^3+48x^2+22x^3-154x^2+132x+12x^2-84x+72 } = \frac{x-4}{8x^4-34x^3-94x^2+48x+72} \end{aligned} $$

Multiply $ \dfrac{8x^2+6x}{x^2-7x+6} $ by $ \dfrac{x-4}{8x^4-34x^3-94x^2+48x+72} $ to get $ \dfrac{x^2-4x}{x^5-12x^4+33x^3+38x^2-132x+72} $.

Step 1: Factor numerators and denominators.

Step 2: Cancel common factors.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{8x^2+6x}{x^2-7x+6} \cdot \frac{x-4}{8x^4-34x^3-94x^2+48x+72} & \xlongequal{\text{Step 1}} \frac{ x \cdot \color{blue}{ \left( 8x+6 \right) } }{ x^2-7x+6 } \cdot \frac{ x-4 }{ \left( x^3-5x^2-8x+12 \right) \cdot \color{blue}{ \left( 8x+6 \right) } } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ x }{ x^2-7x+6 } \cdot \frac{ x-4 }{ x^3-5x^2-8x+12 } \xlongequal{\text{Step 3}} \frac{ x \cdot \left( x-4 \right) }{ \left( x^2-7x+6 \right) \cdot \left( x^3-5x^2-8x+12 \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ x^2-4x }{ x^5-5x^4-8x^3+12x^2-7x^4+35x^3+56x^2-84x+6x^3-30x^2-48x+72 } = \frac{x^2-4x}{x^5-12x^4+33x^3+38x^2-132x+72} \end{aligned} $$