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

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

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

Subtract $ \dfrac{4}{x^2} $ from $ \dfrac{x^4+x}{x^2} $ to get $ \dfrac{ x^4+x - 4 }{ \color{blue}{ x^2 }}$.

To subtract expressions with the same denominators, we subtract the numerators and write the result over the common denominator.

$$ \begin{aligned} \frac{x^4+x}{x^2} - \frac{4}{x^2} & = \frac{x^4+x}{\color{blue}{x^2}} - \frac{4}{\color{blue}{x^2}} =\frac{ x^4+x - 4 }{ \color{blue}{ x^2 }} \end{aligned} $$

Subtract $ \dfrac{1}{x^2} $ from $ \dfrac{x^4+x-4}{x^2} $ to get $ \dfrac{x^4+x-5}{x^2} $.

To subtract expressions with the same denominators, we subtract the numerators and write the result over the common denominator.

$$ \begin{aligned} \frac{x^4+x-4}{x^2} - \frac{1}{x^2} & = \frac{x^4+x-4}{\color{blue}{x^2}} - \frac{1}{\color{blue}{x^2}} = \\[1ex] &=\frac{ x^4+x-4 - 1 }{ \color{blue}{ x^2 }}= \frac{x^4+x-5}{x^2} \end{aligned} $$

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

Step 1: Write $ 5x $ 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 $\color{blue}{x^2}$.

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

Add $ \dfrac{x^4+5x^3+x-5}{x^2} $ and $ 6 $ to get $ \dfrac{ \color{purple}{ x^4+5x^3+6x^2+x-5 } }{ x^2 }$.

Step 1: Write $ 6 $ 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 $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{x^4+5x^3+x-5}{x^2} +6 & \xlongequal{\text{Step 1}} \frac{x^4+5x^3+x-5}{x^2} + \frac{6}{\color{red}{1}} = \frac{ x^4+5x^3+x-5 }{ x^2 } + \frac{ 6 \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^4+5x^3+x-5 } }{ x^2 } + \frac{ \color{purple}{ 6x^2 } }{ x^2 }=\frac{ \color{purple}{ x^4+5x^3+6x^2+x-5 } }{ x^2 } \end{aligned} $$