Find $ \left(x+1\right)^2 $ using formula.

$$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$

where $ A = \color{blue}{ x } $ and $ B = \color{red}{ 1 }$.

$$ \begin{aligned}\left(x+1\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 1 + \color{red}{1^2} = x^2+2x+1\end{aligned} $$

Add $ \dfrac{x}{x^2+2x+1} $ and $ \dfrac{4}{x+1} $ to get $ \dfrac{ \color{purple}{ 5x+4 } }{ 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 $\color{blue}{x+1}$.

$$ \begin{aligned} \frac{x}{x^2+2x+1} + \frac{4}{x+1} & = \frac{ x }{ x^2+2x+1 } + \frac{ 4 \cdot \color{blue}{ \left( x+1 \right) }}{ \left( x+1 \right) \cdot \color{blue}{ \left( x+1 \right) }} = \\[1ex] &=\frac{ \color{purple}{ x } }{ x^2+2x+1 } + \frac{ \color{purple}{ 4x+4 } }{ x^2+2x+1 }=\frac{ \color{purple}{ 5x+4 } }{ x^2+2x+1 } \end{aligned} $$