Simplify $ \dfrac{2x-2}{x^2-1} $ to $ \dfrac{2}{x+1} $.

Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{x-1}$.

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

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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{2x}{x+1} $ and $ 1 $ to get $ \dfrac{ \color{purple}{ 3x+1 } }{ x+1 }$.

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

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