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

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

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

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

To add expressions with the same denominators, we add the numerators and write the result over the common denominator.

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

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

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

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