Multiply $ \color{blue}{x} $ by $ \left( x-1\right) $ $$ \color{blue}{x} \cdot \left( x-1\right) = x^2-x $$

Multiply $x$ by $ \dfrac{x-1}{2} $ to get $ \dfrac{ x^2-x }{ 2 } $.

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.

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

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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $ \dfrac{x^3-3x^2+2x}{6} $ by $ 12 $ to get $ \dfrac{ 12x^3-36x^2+24x }{ 6 } $.

Step 1: Write $ 12 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{x^3-3x^2+2x}{6} \cdot 12 & \xlongequal{\text{Step 1}} \frac{x^3-3x^2+2x}{6} \cdot \frac{12}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( x^3-3x^2+2x \right) \cdot 12 }{ 6 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 12x^3-36x^2+24x }{ 6 } \end{aligned} $$

Add $1+x$ and $ \dfrac{-2x^2+2x}{2} $ to get $ \dfrac{ \color{purple}{ -2x^2+4x+2 } }{ 2 }$.

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

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

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

Multiply $ \dfrac{x^3-3x^2+2x}{6} $ by $ 12 $ to get $ \dfrac{ 12x^3-36x^2+24x }{ 6 } $.

Step 1: Write $ 12 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{x^3-3x^2+2x}{6} \cdot 12 & \xlongequal{\text{Step 1}} \frac{x^3-3x^2+2x}{6} \cdot \frac{12}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( x^3-3x^2+2x \right) \cdot 12 }{ 6 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 12x^3-36x^2+24x }{ 6 } \end{aligned} $$

Add $ \dfrac{-2x^2+4x+2}{2} $ and $ \dfrac{12x^3-36x^2+24x}{6} $ to get $ \dfrac{ \color{purple}{ 12x^3-42x^2+36x+6 } }{ 6 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 3 }$.

$$ \begin{aligned} \frac{-2x^2+4x+2}{2} + \frac{12x^3-36x^2+24x}{6} & = \frac{ \left( -2x^2+4x+2 \right) \cdot \color{blue}{ 3 }}{ 2 \cdot \color{blue}{ 3 }} + \frac{ 12x^3-36x^2+24x }{ 6 } = \\[1ex] &=\frac{ \color{purple}{ -6x^2+12x+6 } }{ 6 } + \frac{ \color{purple}{ 12x^3-36x^2+24x } }{ 6 }=\frac{ \color{purple}{ 12x^3-42x^2+36x+6 } }{ 6 } \end{aligned} $$