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

$$ \begin{aligned} \frac{1}{2} \cdot x & \xlongequal{\text{Step 1}} \frac{1}{2} \cdot \frac{x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot x }{ 2 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x }{ 2 } \end{aligned} $$

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.

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

Multiply $ \dfrac{1}{4} $ by $ x $ to get $ \dfrac{ x }{ 4 } $.

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.

$$ \begin{aligned} \frac{1}{4} \cdot x & \xlongequal{\text{Step 1}} \frac{1}{4} \cdot \frac{x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot x }{ 4 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x }{ 4 } \end{aligned} $$

Subtract $ \dfrac{x}{2} $ from $ \dfrac{1}{4} $ to get $ \dfrac{ \color{purple}{ -2x+1 } }{ 4 }$.

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

$$ \begin{aligned} \frac{1}{4} - \frac{x}{2} & = \frac{ 1 }{ 4 } - \frac{ x \cdot \color{blue}{ 2 }}{ 2 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ 1 } }{ 4 } - \frac{ \color{purple}{ 2x } }{ 4 }=\frac{ \color{purple}{ -2x+1 } }{ 4 } \end{aligned} $$

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

Step 1: Write $ 1 $ 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 $\color{blue}{ 2 }$.

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

Multiply $ \dfrac{1}{4} $ by $ x $ to get $ \dfrac{ x }{ 4 } $.

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.

$$ \begin{aligned} \frac{1}{4} \cdot x & \xlongequal{\text{Step 1}} \frac{1}{4} \cdot \frac{x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot x }{ 4 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x }{ 4 } \end{aligned} $$

Multiply $x^3+1$ by $ \dfrac{-2x+1}{4} $ to get $ \dfrac{ -2x^4+x^3-2x+1 }{ 4 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Subtract $ \dfrac{x}{4} $ from $ \dfrac{x^2+2}{2} $ to get $ \dfrac{ \color{purple}{ 2x^2-x+4 } }{ 4 }$.

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

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

Multiply $x^3+1$ by $ \dfrac{-2x+1}{4} $ to get $ \dfrac{ -2x^4+x^3-2x+1 }{ 4 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $x^2+1$ by $ \dfrac{2x^2-x+4}{4} $ to get $ \dfrac{2x^4-x^3+6x^2-x+4}{4} $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{-2x^4+x^3-2x+1}{4} $ and $ \dfrac{2x^4-x^3+6x^2-x+4}{4} $ to get $ \dfrac{6x^2-3x+5}{4} $.

To add expressions with the same denominators, we add the numerators and write the result over the common denominator.

$$ \begin{aligned} \frac{-2x^4+x^3-2x+1}{4} + \frac{2x^4-x^3+6x^2-x+4}{4} & = \frac{-2x^4+x^3-2x+1}{\color{blue}{4}} + \frac{2x^4-x^3+6x^2-x+4}{\color{blue}{4}} = \\[1ex] &=\frac{ -2x^4+x^3-2x+1 + \left( 2x^4-x^3+6x^2-x+4 \right) }{ \color{blue}{ 4 }}= \frac{6x^2-3x+5}{4} \end{aligned} $$