Subtract $ \dfrac{1}{r+2} $ from $ \dfrac{r+6}{r} $ to get $ \dfrac{ \color{purple}{ r^2+7r+12 } }{ r^2+2r }$.

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

$$ \begin{aligned} \frac{r+6}{r} - \frac{1}{r+2} & = \frac{ \left( r+6 \right) \cdot \color{blue}{ \left( r+2 \right) }}{ r \cdot \color{blue}{ \left( r+2 \right) }} - \frac{ 1 \cdot \color{blue}{ r }}{ \left( r+2 \right) \cdot \color{blue}{ r }} = \\[1ex] &=\frac{ \color{purple}{ r^2+2r+6r+12 } }{ r^2+2r } - \frac{ \color{purple}{ r } }{ r^2+2r }=\frac{ \color{purple}{ r^2+7r+12 } }{ r^2+2r } \end{aligned} $$