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{8}{15} $ by $ x-2 $ to get $ \dfrac{ 8x-16 }{ 15 } $.

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.

$$ \begin{aligned} \frac{8}{15} \cdot x-2 & \xlongequal{\text{Step 1}} \frac{8}{15} \cdot \frac{x-2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 8 \cdot \left( x-2 \right) }{ 15 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 8x-16 }{ 15 } \end{aligned} $$

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{8x-16}{15} $ from $ \dfrac{x-2}{2} $ to get $ \dfrac{ \color{purple}{ -x+2 } }{ 30 }$.

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}{ 15 }$ and the second by $\color{blue}{ 2 }$.

$$ \begin{aligned} \frac{x-2}{2} - \frac{8x-16}{15} & = \frac{ \left( x-2 \right) \cdot \color{blue}{ 15 }}{ 2 \cdot \color{blue}{ 15 }} - \frac{ \left( 8x-16 \right) \cdot \color{blue}{ 2 }}{ 15 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ 15x-30 } }{ 30 } - \frac{ \color{purple}{ 16x-32 } }{ 30 }=\frac{ \color{purple}{ 15x-30 - \left( 16x-32 \right) } }{ 30 } = \\[1ex] &=\frac{ \color{purple}{ -x+2 } }{ 30 } \end{aligned} $$

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{-x+2}{30} $ by $ x-4 $ to get $ \dfrac{-x^2+6x-8}{30} $.

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.

$$ \begin{aligned} \frac{-x+2}{30} \cdot x-4 & \xlongequal{\text{Step 1}} \frac{-x+2}{30} \cdot \frac{x-4}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( -x+2 \right) \cdot \left( x-4 \right) }{ 30 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ -x^2+4x+2x-8 }{ 30 } = \frac{-x^2+6x-8}{30} \end{aligned} $$

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{-x^2+6x-8}{30} $ and $ \dfrac{x^2-4x+4}{12} $ to get $ \dfrac{ \color{purple}{ 3x^2-8x+4 } }{ 60 }$.

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 }$ and the second by $\color{blue}{ 5 }$.

$$ \begin{aligned} \frac{-x^2+6x-8}{30} + \frac{x^2-4x+4}{12} & = \frac{ \left( -x^2+6x-8 \right) \cdot \color{blue}{ 2 }}{ 30 \cdot \color{blue}{ 2 }} + \frac{ \left( x^2-4x+4 \right) \cdot \color{blue}{ 5 }}{ 12 \cdot \color{blue}{ 5 }} = \\[1ex] &=\frac{ \color{purple}{ -2x^2+12x-16 } }{ 60 } + \frac{ \color{purple}{ 5x^2-20x+20 } }{ 60 }=\frac{ \color{purple}{ 3x^2-8x+4 } }{ 60 } \end{aligned} $$