Add $ \dfrac{-x^3}{4} $ and $ 10x^5 $ to get $ \dfrac{ \color{purple}{ 40x^5-x^3 } }{ 4 }$.

Step 1: Write $ 10x^5 $ 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 $\color{blue}{4}$.

$$ \begin{aligned} \frac{-x^3}{4} +10x^5 & \xlongequal{\text{Step 1}} \frac{-x^3}{4} + \frac{10x^5}{\color{red}{1}} = \frac{ -x^3 }{ 4 } + \frac{ 10x^5 \cdot \color{blue}{ 4 }}{ 1 \cdot \color{blue}{ 4 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -x^3 } }{ 4 } + \frac{ \color{purple}{ 40x^5 } }{ 4 }=\frac{ \color{purple}{ 40x^5-x^3 } }{ 4 } \end{aligned} $$

Subtract $5$ from $ \dfrac{40x^5-x^3}{4} $ to get $ \dfrac{ \color{purple}{ 40x^5-x^3-20 } }{ 4 }$.

Step 1: Write $ 5 $ 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 $\color{blue}{4}$.

$$ \begin{aligned} \frac{40x^5-x^3}{4} -5 & \xlongequal{\text{Step 1}} \frac{40x^5-x^3}{4} - \frac{5}{\color{red}{1}} = \frac{ 40x^5-x^3 }{ 4 } - \frac{ 5 \cdot \color{blue}{ 4 }}{ 1 \cdot \color{blue}{ 4 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 40x^5-x^3 } }{ 4 } - \frac{ \color{purple}{ 20 } }{ 4 }=\frac{ \color{purple}{ 40x^5-x^3-20 } }{ 4 } \end{aligned} $$