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.

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

Subtract $3$ from $ \dfrac{x^2}{2} $ to get $ \dfrac{ \color{purple}{ x^2-6 } }{ 2 }$.

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

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

Multiply $ \dfrac{x^2-6}{2} $ by $ 4y^3+5x^2 $ to get $ \dfrac{ 4x^2y^3+5x^4-24y^3-30x^2 }{ 2 } $.

Step 1: Write $ 4y^3+5x^2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{x^2-6}{2} \cdot 4y^3+5x^2 & \xlongequal{\text{Step 1}} \frac{x^2-6}{2} \cdot \frac{4y^3+5x^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( x^2-6 \right) \cdot \left( 4y^3+5x^2 \right) }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4x^2y^3+5x^4-24y^3-30x^2 }{ 2 } \end{aligned} $$