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+1 $ to get $ 2$.

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

Step 2: Cancel $ \color{red}{ x+1 } $ in first and second fraction.

Step 3: Multiply numerators and denominators.

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