Simplify $ \dfrac{8n-72}{54+3n-n^2} $ to $ \dfrac{8}{-n-6} $.

Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{n-9}$.

$$ \begin{aligned} \frac{8n-72}{54+3n-n^2} & =\frac{ 8 \cdot \color{blue}{ \left( n-9 \right) }}{ \left( -n-6 \right) \cdot \color{blue}{ \left( n-9 \right) }} = \\[1ex] &= \frac{8}{-n-6} \end{aligned} $$

Simplify $ \dfrac{n^2+9n+18}{5n+15} $ to $ \dfrac{n+6}{5} $.

Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{n+3}$.

$$ \begin{aligned} \frac{n^2+9n+18}{5n+15} & =\frac{ \left( n+6 \right) \cdot \color{blue}{ \left( n+3 \right) }}{ 5 \cdot \color{blue}{ \left( n+3 \right) }} = \\[1ex] &= \frac{n+6}{5} \end{aligned} $$

Multiply $ \dfrac{8}{-n-6} $ by $ \dfrac{n+6}{5} $ to get $ \dfrac{ 8 }{ -5 } $.

Step 1: Factor numerators and denominators.

Step 2: Cancel common factors.

Step 3: Multiply numerators and denominators.

$$ \begin{aligned} \frac{8}{-n-6} \cdot \frac{n+6}{5} & \xlongequal{\text{Step 1}} \frac{ 8 }{ \left( -1 \right) \cdot \color{red}{ \left( n+6 \right) } } \cdot \frac{ 1 \cdot \color{red}{ \left( n+6 \right) } }{ 5 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 8 }{ -1 } \cdot \frac{ 1 }{ 5 } \xlongequal{\text{Step 3}} \frac{ 8 }{ -5 } \end{aligned} $$

Place minus sign in front of the fraction.