Use the distributive property.

Multiply $ \dfrac{1}{12} $ by $ x-2 $ to get $ \dfrac{ x-2 }{ 12 } $.

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.

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

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

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.

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

Multiply $ \dfrac{10}{3} $ by $ x-2 $ to get $ \dfrac{ 10x-20 }{ 3 } $.

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.

$$ \begin{aligned} \frac{10}{3} \cdot x-2 & \xlongequal{\text{Step 1}} \frac{10}{3} \cdot \frac{x-2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 10 \cdot \left( x-2 \right) }{ 3 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 10x-20 }{ 3 } \end{aligned} $$

Multiply $ \dfrac{x-2}{12} $ by $ x-2 $ to get $ \dfrac{x^2-4x+4}{12} $.

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.

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

Subtract $ \dfrac{10x-20}{3} $ from $ \dfrac{x-2}{2} $ to get $ \dfrac{ \color{purple}{ -17x+34 } }{ 6 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ 3 }$ and the second by $\color{blue}{ 2 }$.

$$ \begin{aligned} \frac{x-2}{2} - \frac{10x-20}{3} & = \frac{ \left( x-2 \right) \cdot \color{blue}{ 3 }}{ 2 \cdot \color{blue}{ 3 }} - \frac{ \left( 10x-20 \right) \cdot \color{blue}{ 2 }}{ 3 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ 3x-6 } }{ 6 } - \frac{ \color{purple}{ 20x-40 } }{ 6 }=\frac{ \color{purple}{ 3x-6 - \left( 20x-40 \right) } }{ 6 } = \\[1ex] &=\frac{ \color{purple}{ -17x+34 } }{ 6 } \end{aligned} $$

Multiply $ \dfrac{x-2}{12} $ by $ x-2 $ to get $ \dfrac{x^2-4x+4}{12} $.

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.

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

Multiply $ \dfrac{-17x+34}{6} $ by $ x-4 $ to get $ \dfrac{-17x^2+102x-136}{6} $.

Step 1: Write $ x-4 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{-17x+34}{6} \cdot x-4 & \xlongequal{\text{Step 1}} \frac{-17x+34}{6} \cdot \frac{x-4}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( -17x+34 \right) \cdot \left( x-4 \right) }{ 6 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ -17x^2+68x+34x-136 }{ 6 } = \frac{-17x^2+102x-136}{6} \end{aligned} $$

Multiply $ \dfrac{x-2}{12} $ by $ x-2 $ to get $ \dfrac{x^2-4x+4}{12} $.

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.

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

Add $ \dfrac{-17x^2+102x-136}{6} $ and $ \dfrac{x^2-4x+4}{12} $ to get $ \dfrac{ \color{purple}{ -33x^2+200x-268 } }{ 12 }$.

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

$$ \begin{aligned} \frac{-17x^2+102x-136}{6} + \frac{x^2-4x+4}{12} & = \frac{ \left( -17x^2+102x-136 \right) \cdot \color{blue}{ 2 }}{ 6 \cdot \color{blue}{ 2 }} + \frac{ x^2-4x+4 }{ 12 } = \\[1ex] &=\frac{ \color{purple}{ -34x^2+204x-272 } }{ 12 } + \frac{ \color{purple}{ x^2-4x+4 } }{ 12 }=\frac{ \color{purple}{ -33x^2+200x-268 } }{ 12 } \end{aligned} $$