Divide $ \dfrac{4}{x} $ by $ x^2 $ to get $ \dfrac{ 4 }{ x^3 } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

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

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}{ x^3 }$ and the second by $\color{blue}{ x }$.

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

Add $ \dfrac{x^6+4x}{x^4} $ and $ 8x $ to get $ \dfrac{ \color{purple}{ x^6+8x^5+4x } }{ x^4 }$.

Step 1: Write $ 8x $ 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 $\color{blue}{x^4}$.

$$ \begin{aligned} \frac{x^6+4x}{x^4} +8x & \xlongequal{\text{Step 1}} \frac{x^6+4x}{x^4} + \frac{8x}{\color{red}{1}} = \frac{ x^6+4x }{ x^4 } + \frac{ 8x \cdot \color{blue}{ x^4 }}{ 1 \cdot \color{blue}{ x^4 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^6+4x } }{ x^4 } + \frac{ \color{purple}{ 8x^5 } }{ x^4 }=\frac{ \color{purple}{ x^6+8x^5+4x } }{ x^4 } \end{aligned} $$

Add $ \dfrac{x^6+8x^5+4x}{x^4} $ and $ 16 $ to get $ \dfrac{ \color{purple}{ x^6+8x^5+16x^4+4x } }{ x^4 }$.

Step 1: Write $ 16 $ 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 $\color{blue}{x^4}$.

$$ \begin{aligned} \frac{x^6+8x^5+4x}{x^4} +16 & \xlongequal{\text{Step 1}} \frac{x^6+8x^5+4x}{x^4} + \frac{16}{\color{red}{1}} = \frac{ x^6+8x^5+4x }{ x^4 } + \frac{ 16 \cdot \color{blue}{ x^4 }}{ 1 \cdot \color{blue}{ x^4 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^6+8x^5+4x } }{ x^4 } + \frac{ \color{purple}{ 16x^4 } }{ x^4 }=\frac{ \color{purple}{ x^6+8x^5+16x^4+4x } }{ x^4 } \end{aligned} $$