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

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

Add $3xy$ and $ \dfrac{x^4}{2} $ to get $ \dfrac{ \color{purple}{ x^4+6xy } }{ 2 }$.

Step 1: Write $ 3xy $ 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 $\color{blue}{ 2 }$.

$$ \begin{aligned} 3xy+ \frac{x^4}{2} & \xlongequal{\text{Step 1}} \frac{3xy}{\color{red}{1}} + \frac{x^4}{2} = \frac{ 3xy \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} + \frac{ x^4 }{ 2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 6xy } }{ 2 } + \frac{ \color{purple}{ x^4 } }{ 2 }=\frac{ \color{purple}{ x^4+6xy } }{ 2 } \end{aligned} $$

Add $ \dfrac{x^4+6xy}{2} $ and $ 5x $ to get $ \dfrac{ \color{purple}{ x^4+6xy+10x } }{ 2 }$.

Step 1: Write $ 5x $ 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 $\color{blue}{2}$.

$$ \begin{aligned} \frac{x^4+6xy}{2} +5x & \xlongequal{\text{Step 1}} \frac{x^4+6xy}{2} + \frac{5x}{\color{red}{1}} = \frac{ x^4+6xy }{ 2 } + \frac{ 5x \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^4+6xy } }{ 2 } + \frac{ \color{purple}{ 10x } }{ 2 }=\frac{ \color{purple}{ x^4+6xy+10x } }{ 2 } \end{aligned} $$