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

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

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

Add $ \dfrac{-x^2+2x-1}{x^3+x} $ and $ \dfrac{1}{x^3+x} $ to get $ \dfrac{-x^2+2x}{x^3+x} $.

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

$$ \begin{aligned} \frac{-x^2+2x-1}{x^3+x} + \frac{1}{x^3+x} & = \frac{-x^2+2x-1}{\color{blue}{x^3+x}} + \frac{1}{\color{blue}{x^3+x}} = \\[1ex] &=\frac{ -x^2+2x-1 + 1 }{ \color{blue}{ x^3+x }}= \frac{-x^2+2x}{x^3+x} \end{aligned} $$