Multiply $ \dfrac{1}{8} $ by $ x^3-2x^2-5x+6 $ to get $ \dfrac{ x^3-2x^2-5x+6 }{ 8 } $.

Step 1: Write $ x^3-2x^2-5x+6 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $ \dfrac{1}{16} $ by $ x-5 $ to get $ \dfrac{ x-5 }{ 16 } $.

Step 1: Write $ x-5 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{16} \cdot x-5 & \xlongequal{\text{Step 1}} \frac{1}{16} \cdot \frac{x-5}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot \left( x-5 \right) }{ 16 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x-5 }{ 16 } \end{aligned} $$

Multiply $ \dfrac{1}{8} $ by $ x^3-2x^2-5x+6 $ to get $ \dfrac{ x^3-2x^2-5x+6 }{ 8 } $.

Step 1: Write $ x^3-2x^2-5x+6 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $ \dfrac{x-5}{16} $ by $ x-2 $ to get $ \dfrac{x^2-7x+10}{16} $.

Step 1: Write $ x-2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{x-5}{16} \cdot x-2 & \xlongequal{\text{Step 1}} \frac{x-5}{16} \cdot \frac{x-2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( x-5 \right) \cdot \left( x-2 \right) }{ 16 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^2-2x-5x+10 }{ 16 } = \frac{x^2-7x+10}{16} \end{aligned} $$

Multiply $ \dfrac{1}{8} $ by $ x^3-2x^2-5x+6 $ to get $ \dfrac{ x^3-2x^2-5x+6 }{ 8 } $.

Step 1: Write $ x^3-2x^2-5x+6 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $ \dfrac{x^2-7x+10}{16} $ by $ x^3+2x^2-x-2 $ to get $ \dfrac{x^5-5x^4-5x^3+25x^2+4x-20}{16} $.

Step 1: Write $ x^3+2x^2-x-2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{x^2-7x+10}{16} \cdot x^3+2x^2-x-2 & \xlongequal{\text{Step 1}} \frac{x^2-7x+10}{16} \cdot \frac{x^3+2x^2-x-2}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^2-7x+10 \right) \cdot \left( x^3+2x^2-x-2 \right) }{ 16 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x^5+2x^4-x^3-2x^2-7x^4-14x^3+7x^2+14x+10x^3+20x^2-10x-20 }{ 16 } = \\[1ex] &= \frac{x^5-5x^4-5x^3+25x^2+4x-20}{16} \end{aligned} $$

Add $ \dfrac{x^3-2x^2-5x+6}{8} $ and $ \dfrac{x^5-5x^4-5x^3+25x^2+4x-20}{16} $ to get $ \dfrac{ \color{purple}{ x^5-5x^4-3x^3+21x^2-6x-8 } }{ 16 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 2 }$.

$$ \begin{aligned} \frac{x^3-2x^2-5x+6}{8} + \frac{x^5-5x^4-5x^3+25x^2+4x-20}{16} & = \frac{ \left( x^3-2x^2-5x+6 \right) \cdot \color{blue}{ 2 }}{ 8 \cdot \color{blue}{ 2 }} + \frac{ x^5-5x^4-5x^3+25x^2+4x-20 }{ 16 } = \\[1ex] &=\frac{ \color{purple}{ 2x^3-4x^2-10x+12 } }{ 16 } + \frac{ \color{purple}{ x^5-5x^4-5x^3+25x^2+4x-20 } }{ 16 } = \\[1ex] &=\frac{ \color{purple}{ x^5-5x^4-3x^3+21x^2-6x-8 } }{ 16 } \end{aligned} $$