Simplify $ \dfrac{1-x^2}{x^2+8x-9} $ to $ \dfrac{-x-1}{x+9} $.

Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{x-1}$.

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

Divide $ \dfrac{-x-1}{x+9} $ by $ x^2-4x-12 $ to get $ \dfrac{-x-1}{x^3+5x^2-48x-108} $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{-x-1}{x+9} }{x^2-4x-12} & \xlongequal{\text{Step 1}} \frac{-x-1}{x+9} \cdot \frac{\color{blue}{1}}{\color{blue}{x^2-4x-12}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( -x-1 \right) \cdot 1 }{ \left( x+9 \right) \cdot \left( x^2-4x-12 \right) } \xlongequal{\text{Step 3}} \frac{ -x-1 }{ x^3-4x^2-12x+9x^2-36x-108 } = \\[1ex] &= \frac{-x-1}{x^3+5x^2-48x-108} \end{aligned} $$

Divide $ \dfrac{-x-1}{x^3+5x^2-48x-108} $ by $ 36-x^2 $ to get $ \dfrac{-x-1}{-x^5-5x^4+84x^3+288x^2-1728x-3888} $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{-x-1}{x^3+5x^2-48x-108} }{36-x^2} & \xlongequal{\text{Step 1}} \frac{-x-1}{x^3+5x^2-48x-108} \cdot \frac{\color{blue}{1}}{\color{blue}{36-x^2}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( -x-1 \right) \cdot 1 }{ \left( x^3+5x^2-48x-108 \right) \cdot \left( 36-x^2 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ -x-1 }{ 36x^3-x^5+180x^2-5x^4-1728x+48x^3-3888+108x^2 } = \frac{-x-1}{-x^5-5x^4+84x^3+288x^2-1728x-3888} \end{aligned} $$

Multiply $x$ by $ \dfrac{-x-1}{-x^5-5x^4+84x^3+288x^2-1728x-3888} $ to get $ \dfrac{ -x^2-x }{ -x^5-5x^4+84x^3+288x^2-1728x-3888 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} x \cdot \frac{-x-1}{-x^5-5x^4+84x^3+288x^2-1728x-3888} & \xlongequal{\text{Step 1}} \frac{x}{\color{red}{1}} \cdot \frac{-x-1}{-x^5-5x^4+84x^3+288x^2-1728x-3888} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ x \cdot \left( -x-1 \right) }{ 1 \cdot \left( -x^5-5x^4+84x^3+288x^2-1728x-3888 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ -x^2-x }{ -x^5-5x^4+84x^3+288x^2-1728x-3888 } \end{aligned} $$

Add $ \dfrac{x^2+11x+18}{x^2} $ and $ \dfrac{-x^2-x}{-x^5-5x^4+84x^3+288x^2-1728x-3888} $ to get $ \dfrac{ \color{purple}{ -x^7-16x^6+11x^5+1121x^4+2951x^3-17712x^2-73872x-69984 } }{ -x^7-5x^6+84x^5+288x^4-1728x^3-3888x^2 }$.

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^5-5x^4+84x^3+288x^2-1728x-3888 }$ and the second by $\color{blue}{ x^2 }$.

$$ \begin{aligned} \frac{x^2+11x+18}{x^2} + \frac{-x^2-x}{-x^5-5x^4+84x^3+288x^2-1728x-3888} & = \frac{ \left( x^2+11x+18 \right) \cdot \color{blue}{ \left( -x^5-5x^4+84x^3+288x^2-1728x-3888 \right) }}{ x^2 \cdot \color{blue}{ \left( -x^5-5x^4+84x^3+288x^2-1728x-3888 \right) }} + \frac{ \left( -x^2-x \right) \cdot \color{blue}{ x^2 }}{ \left( -x^5-5x^4+84x^3+288x^2-1728x-3888 \right) \cdot \color{blue}{ x^2 }} = \\[1ex] &=\frac{ \color{purple}{ -x^7-5x^6+84x^5+288x^4-1728x^3-3888x^2-11x^6-55x^5+924x^4+3168x^3-19008x^2-42768x-18x^5-90x^4+1512x^3+5184x^2-31104x-69984 } }{ -x^7-5x^6+84x^5+288x^4-1728x^3-3888x^2 } + \frac{ \color{purple}{ -x^4-x^3 } }{ -x^7-5x^6+84x^5+288x^4-1728x^3-3888x^2 } = \\[1ex] &=\frac{ \color{purple}{ -x^7-16x^6+11x^5+1121x^4+2951x^3-17712x^2-73872x-69984 } }{ -x^7-5x^6+84x^5+288x^4-1728x^3-3888x^2 } \end{aligned} $$