Subtract $ \dfrac{x^5}{2} $ from $ -6 $ to get $ \dfrac{ \color{purple}{ -x^5-12 } }{ 2 }$.

Step 1: Write $ -6 $ 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 first fraction by $\color{blue}{ 2 }$.

$$ \begin{aligned} -6- \frac{x^5}{2} & \xlongequal{\text{Step 1}} \frac{-6}{\color{red}{1}} - \frac{x^5}{2} = \frac{ \left( -6 \right) \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} - \frac{ x^5 }{ 2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -12 } }{ 2 } - \frac{ \color{purple}{ x^5 } }{ 2 }=\frac{ \color{purple}{ -x^5-12 } }{ 2 } \end{aligned} $$

Subtract $x^2$ from $ \dfrac{-x^5-12}{2} $ to get $ \dfrac{ \color{purple}{ -x^5-2x^2-12 } }{ 2 }$.

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

$$ \begin{aligned} \frac{-x^5-12}{2} -x^2 & \xlongequal{\text{Step 1}} \frac{-x^5-12}{2} - \frac{x^2}{\color{red}{1}} = \frac{ -x^5-12 }{ 2 } - \frac{ x^2 \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -x^5-12 } }{ 2 } - \frac{ \color{purple}{ 2x^2 } }{ 2 }=\frac{ \color{purple}{ -x^5-2x^2-12 } }{ 2 } \end{aligned} $$