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 $ y^2 $ to get $ \dfrac{ y^2 }{ 4 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{4} \cdot y^2 & \xlongequal{\text{Step 1}} \frac{1}{4} \cdot \frac{y^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot y^2 }{ 4 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ y^2 }{ 4 } \end{aligned} $$

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

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

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

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

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

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{4}$.

$$ \begin{aligned} \frac{2x^2-y^2}{4} -xy & \xlongequal{\text{Step 1}} \frac{2x^2-y^2}{4} - \frac{xy}{\color{red}{1}} = \frac{ 2x^2-y^2 }{ 4 } - \frac{ xy \cdot \color{blue}{ 4 }}{ 1 \cdot \color{blue}{ 4 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2x^2-y^2 } }{ 4 } - \frac{ \color{purple}{ 4xy } }{ 4 }=\frac{ \color{purple}{ 2x^2-4xy-y^2 } }{ 4 } \end{aligned} $$

Add $ \dfrac{2x^2-4xy-y^2}{4} $ and $ 2x $ to get $ \dfrac{ \color{purple}{ 2x^2-4xy-y^2+8x } }{ 4 }$.

Step 1: Write $ 2x $ 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 second fraction by $\color{blue}{4}$.

$$ \begin{aligned} \frac{2x^2-4xy-y^2}{4} +2x & \xlongequal{\text{Step 1}} \frac{2x^2-4xy-y^2}{4} + \frac{2x}{\color{red}{1}} = \frac{ 2x^2-4xy-y^2 }{ 4 } + \frac{ 2x \cdot \color{blue}{ 4 }}{ 1 \cdot \color{blue}{ 4 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2x^2-4xy-y^2 } }{ 4 } + \frac{ \color{purple}{ 8x } }{ 4 }=\frac{ \color{purple}{ 2x^2-4xy-y^2+8x } }{ 4 } \end{aligned} $$

Subtract $0$ from $ \dfrac{2x^2-4xy-y^2+8x}{4} $ to get $ \dfrac{ \color{purple}{ 2x^2-4xy-y^2+8x } }{ 4 }$.

Step 1: Write $ 0 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{4}$.

$$ \begin{aligned} \frac{2x^2-4xy-y^2+8x}{4} -0 & \xlongequal{\text{Step 1}} \frac{2x^2-4xy-y^2+8x}{4} - \frac{0}{\color{red}{1}} = \frac{ 2x^2-4xy-y^2+8x }{ 4 } - \frac{ 0 \cdot \color{blue}{ 4 }}{ 1 \cdot \color{blue}{ 4 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2x^2-4xy-y^2+8x } }{ 4 } - \frac{ \color{purple}{ 0 } }{ 4 }=\frac{ \color{purple}{ 2x^2-4xy-y^2+8x } }{ 4 } \end{aligned} $$