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

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{x+1}$.

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