Add $ \dfrac{r}{r^2+11r+6} $ and $ r $ to get $ \dfrac{ \color{purple}{ r^3+11r^2+7r } }{ r^2+11r+6 }$.

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

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{r^2+11r+6}$.

$$ \begin{aligned} \frac{r}{r^2+11r+6} +r & \xlongequal{\text{Step 1}} \frac{r}{r^2+11r+6} + \frac{r}{\color{red}{1}} = \frac{ r }{ r^2+11r+6 } + \frac{ r \cdot \color{blue}{ \left( r^2+11r+6 \right) }}{ 1 \cdot \color{blue}{ \left( r^2+11r+6 \right) }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ r } }{ r^2+11r+6 } + \frac{ \color{purple}{ r^3+11r^2+6r } }{ r^2+11r+6 }=\frac{ \color{purple}{ r^3+11r^2+7r } }{ r^2+11r+6 } \end{aligned} $$

Subtract $8$ from $ \dfrac{r^3+11r^2+7r}{r^2+11r+6} $ to get $ \dfrac{ \color{purple}{ r^3+3r^2-81r-48 } }{ r^2+11r+6 }$.

Step 1: Write $ 8 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{r^2+11r+6}$.

$$ \begin{aligned} \frac{r^3+11r^2+7r}{r^2+11r+6} -8 & \xlongequal{\text{Step 1}} \frac{r^3+11r^2+7r}{r^2+11r+6} - \frac{8}{\color{red}{1}} = \frac{ r^3+11r^2+7r }{ r^2+11r+6 } - \frac{ 8 \cdot \color{blue}{ \left( r^2+11r+6 \right) }}{ 1 \cdot \color{blue}{ \left( r^2+11r+6 \right) }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ r^3+11r^2+7r } }{ r^2+11r+6 } - \frac{ \color{purple}{ 8r^2+88r+48 } }{ r^2+11r+6 } = \\[1ex] &=\frac{ \color{purple}{ r^3+11r^2+7r - \left( 8r^2+88r+48 \right) } }{ r^2+11r+6 }=\frac{ \color{purple}{ r^3+3r^2-81r-48 } }{ r^2+11r+6 } \end{aligned} $$