Multiply $6$ by $ \dfrac{x}{x+5} $ to get $ \dfrac{ 6x }{ x+5 } $.

Step 1: Write $ 6 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 6 \cdot \frac{x}{x+5} & \xlongequal{\text{Step 1}} \frac{6}{\color{red}{1}} \cdot \frac{x}{x+5} \xlongequal{\text{Step 2}} \frac{ 6 \cdot x }{ 1 \cdot \left( x+5 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 6x }{ x+5 } \end{aligned} $$

Add $ \dfrac{x}{x+1} $ and $ \dfrac{6x}{x+5} $ to get $ \dfrac{ \color{purple}{ 7x^2+11x } }{ x^2+6x+5 }$.

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+5 }$ and the second by $\color{blue}{ x+1 }$.

$$ \begin{aligned} \frac{x}{x+1} + \frac{6x}{x+5} & = \frac{ x \cdot \color{blue}{ \left( x+5 \right) }}{ \left( x+1 \right) \cdot \color{blue}{ \left( x+5 \right) }} + \frac{ 6x \cdot \color{blue}{ \left( x+1 \right) }}{ \left( x+5 \right) \cdot \color{blue}{ \left( x+1 \right) }} = \\[1ex] &=\frac{ \color{purple}{ x^2+5x } }{ x^2+5x+x+5 } + \frac{ \color{purple}{ 6x^2+6x } }{ x^2+5x+x+5 }=\frac{ \color{purple}{ 7x^2+11x } }{ x^2+6x+5 } \end{aligned} $$