Subtract $ \dfrac{1}{x+1} $ from $ \dfrac{1}{x} $ to get $ \dfrac{ \color{purple}{ 1 } }{ x^2+x }$.

To subtract 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 $\color{blue}{ x }$.

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