Add $2x$ and $ \dfrac{1}{x} $ to get $ \dfrac{ \color{purple}{ 2x^2+1 } }{ x }$.

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

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

Subtract $ \dfrac{1}{x} $ from $ 2x $ to get $ \dfrac{ \color{purple}{ 2x^2-1 } }{ x }$.

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

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

Multiply $ \dfrac{2x^2+1}{x} $ by $ \dfrac{2x^2-1}{x} $ to get $ \dfrac{4x^4-1}{x^2} $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

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