Multiply $7$ by $ \dfrac{n}{n+1} $ to get $ \dfrac{ 7n }{ n+1 } $.

Step 1: Write $ 7 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 7 \cdot \frac{n}{n+1} & \xlongequal{\text{Step 1}} \frac{7}{\color{red}{1}} \cdot \frac{n}{n+1} \xlongequal{\text{Step 2}} \frac{ 7 \cdot n }{ 1 \cdot \left( n+1 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 7n }{ n+1 } \end{aligned} $$

Add $ \dfrac{7n}{n+1} $ and $ \dfrac{8}{n-7} $ to get $ \dfrac{ \color{purple}{ 7n^2-41n+8 } }{ n^2-6n-7 }$.

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}{ n-7 }$ and the second by $\color{blue}{ n+1 }$.

$$ \begin{aligned} \frac{7n}{n+1} + \frac{8}{n-7} & = \frac{ 7n \cdot \color{blue}{ \left( n-7 \right) }}{ \left( n+1 \right) \cdot \color{blue}{ \left( n-7 \right) }} + \frac{ 8 \cdot \color{blue}{ \left( n+1 \right) }}{ \left( n-7 \right) \cdot \color{blue}{ \left( n+1 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 7n^2-49n } }{ n^2-7n+n-7 } + \frac{ \color{purple}{ 8n+8 } }{ n^2-7n+n-7 }=\frac{ \color{purple}{ 7n^2-41n+8 } }{ n^2-6n-7 } \end{aligned} $$