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

Step 1: Write $ 2 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{2x}{x+1} $ and $ \dfrac{5}{2x} $ to get $ \dfrac{ \color{purple}{ 4x^2+5x+5 } }{ 2x^2+2x }$.

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}{ 2x }$ and the second by $\color{blue}{ x+1 }$.

$$ \begin{aligned} \frac{2x}{x+1} + \frac{5}{2x} & = \frac{ 2x \cdot \color{blue}{ 2x }}{ \left( x+1 \right) \cdot \color{blue}{ 2x }} + \frac{ 5 \cdot \color{blue}{ \left( x+1 \right) }}{ 2x \cdot \color{blue}{ \left( x+1 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 4x^2 } }{ 2x^2+2x } + \frac{ \color{purple}{ 5x+5 } }{ 2x^2+2x }=\frac{ \color{purple}{ 4x^2+5x+5 } }{ 2x^2+2x } \end{aligned} $$

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

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

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