Multiply each term of $ \left( \color{blue}{x-1}\right) $ by each term in $ \left( x+2\right) $.

$$ \left( \color{blue}{x-1}\right) \cdot \left( x+2\right) = x^2+2x-x-2 $$

Combine like terms:

$$ x^2+ \color{blue}{2x} \color{blue}{-x} -2 = x^2+ \color{blue}{x} -2 $$

Multiply each term of $ \left( \color{blue}{x^2+x-2}\right) $ by each term in $ \left( x-3\right) $.

$$ \left( \color{blue}{x^2+x-2}\right) \cdot \left( x-3\right) = x^3-3x^2+x^2-3x-2x+6 $$

Combine like terms:

$$ x^3 \color{blue}{-3x^2} + \color{blue}{x^2} \color{red}{-3x} \color{red}{-2x} +6 = x^3 \color{blue}{-2x^2} \color{red}{-5x} +6 $$

Multiply each term of $ \left( \color{blue}{x^3-2x^2-5x+6}\right) $ by each term in $ \left( x+3\right) $.

$$ \left( \color{blue}{x^3-2x^2-5x+6}\right) \cdot \left( x+3\right) = x^4+3x^3-2x^3-6x^2-5x^2-15x+6x+18 $$

Combine like terms:

$$ x^4+ \color{blue}{3x^3} \color{blue}{-2x^3} \color{red}{-6x^2} \color{red}{-5x^2} \color{green}{-15x} + \color{green}{6x} +18 = x^4+ \color{blue}{x^3} \color{red}{-11x^2} \color{green}{-9x} +18 $$

Multiply each term of $ \left( \color{blue}{x-1}\right) $ by each term in $ \left( x+2\right) $.

$$ \left( \color{blue}{x-1}\right) \cdot \left( x+2\right) = x^2+2x-x-2 $$

Combine like terms:

$$ x^2+ \color{blue}{2x} \color{blue}{-x} -2 = x^2+ \color{blue}{x} -2 $$

Multiply each term of $ \left( \color{blue}{x^2+x-2}\right) $ by each term in $ \left( x+4\right) $.

$$ \left( \color{blue}{x^2+x-2}\right) \cdot \left( x+4\right) = x^3+4x^2+x^2+4x-2x-8 $$

Combine like terms:

$$ x^3+ \color{blue}{4x^2} + \color{blue}{x^2} + \color{red}{4x} \color{red}{-2x} -8 = x^3+ \color{blue}{5x^2} + \color{red}{2x} -8 $$

Multiply each term of $ \left( \color{blue}{x^3+5x^2+2x-8}\right) $ by each term in $ \left( x-4\right) $.

$$ \left( \color{blue}{x^3+5x^2+2x-8}\right) \cdot \left( x-4\right) = x^4-4x^3+5x^3-20x^2+2x^2-8x-8x+32 $$

Combine like terms:

$$ x^4 \color{blue}{-4x^3} + \color{blue}{5x^3} \color{red}{-20x^2} + \color{red}{2x^2} \color{green}{-8x} \color{green}{-8x} +32 = x^4+ \color{blue}{x^3} \color{red}{-18x^2} \color{green}{-16x} +32 $$

Multiply $x^4+x^3-11x^2-9x+18$ by $ \dfrac{x+5}{x^4+x^3-18x^2-16x+32} $ to get $ \dfrac{ x^3+5x^2-9x-45 }{ x^2-16 } $.

Step 1: Write $ x^4+x^3-11x^2-9x+18 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Factor numerators and denominators.

Step 3: Cancel common factors.

Step 4: Multiply numerators and denominators.

Step 5: Simplify numerator and denominator.

$$ \begin{aligned} x^4+x^3-11x^2-9x+18 \cdot \frac{x+5}{x^4+x^3-18x^2-16x+32} & \xlongequal{\text{Step 1}} \frac{x^4+x^3-11x^2-9x+18}{\color{red}{1}} \cdot \frac{x+5}{x^4+x^3-18x^2-16x+32} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^2-9 \right) \cdot \color{blue}{ \left( x^2+x-2 \right) } }{ 1 } \cdot \frac{ x+5 }{ \left( x^2-16 \right) \cdot \color{blue}{ \left( x^2+x-2 \right) } } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^2-9 }{ 1 } \cdot \frac{ x+5 }{ x^2-16 } \xlongequal{\text{Step 4}} \frac{ \left( x^2-9 \right) \cdot \left( x+5 \right) }{ 1 \cdot \left( x^2-16 \right) } = \\[1ex] & \xlongequal{\text{Step 5}} \frac{ x^3+5x^2-9x-45 }{ x^2-16 } \end{aligned} $$