Multiply $4$ by $ \dfrac{x}{x^3} $ to get $ \dfrac{ 4x }{ x^3 } $.

Step 1: Write $ 4 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{4x}{x^3} $ and $ 7x^2 $ to get $ \dfrac{ \color{purple}{ 7x^5+4x } }{ x^3 }$.

Step 1: Write $ 7x^2 $ 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 $\color{blue}{x^3}$.

$$ \begin{aligned} \frac{4x}{x^3} +7x^2 & \xlongequal{\text{Step 1}} \frac{4x}{x^3} + \frac{7x^2}{\color{red}{1}} = \frac{ 4x }{ x^3 } + \frac{ 7x^2 \cdot \color{blue}{ x^3 }}{ 1 \cdot \color{blue}{ x^3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4x } }{ x^3 } + \frac{ \color{purple}{ 7x^5 } }{ x^3 }=\frac{ \color{purple}{ 7x^5+4x } }{ x^3 } \end{aligned} $$