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

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

$$ \begin{aligned} \frac{2}{x} + \frac{2}{x^2} & = \frac{ 2 \cdot \color{blue}{ x^2 }}{ x \cdot \color{blue}{ x^2 }} + \frac{ 2 \cdot \color{blue}{ x }}{ x^2 \cdot \color{blue}{ x }} = \\[1ex] &=\frac{ \color{purple}{ 2x^2 } }{ x^3 } + \frac{ \color{purple}{ 2x } }{ x^3 }=\frac{ \color{purple}{ 2x^2+2x } }{ x^3 } \end{aligned} $$

Divide $ \dfrac{2x^2+2x}{x^3} $ by $ 1 $ to get $ \dfrac{ 2x^2+2x }{ 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.

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

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

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.

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

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

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

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