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

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

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

Combine like terms:

$$ x^3+ \color{blue}{2x^2} + \color{blue}{x^2} +2x = x^3+ \color{blue}{3x^2} +2x $$

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

$$ \left( \color{blue}{x^3+3x^2+2x}\right) \cdot \left( x+3\right) = x^4+3x^3+3x^3+9x^2+2x^2+6x $$

Combine like terms:

$$ x^4+ \color{blue}{3x^3} + \color{blue}{3x^3} + \color{red}{9x^2} + \color{red}{2x^2} +6x = x^4+ \color{blue}{6x^3} + \color{red}{11x^2} +6x $$

Multiply $x^4+6x^3+11x^2+6x$ by $ \dfrac{x+4}{120} $ to get $ \dfrac{x^5+10x^4+35x^3+50x^2+24x}{120} $.

Step 1: Write $ x^4+6x^3+11x^2+6x $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} x^4+6x^3+11x^2+6x \cdot \frac{x+4}{120} & \xlongequal{\text{Step 1}} \frac{x^4+6x^3+11x^2+6x}{\color{red}{1}} \cdot \frac{x+4}{120} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^4+6x^3+11x^2+6x \right) \cdot \left( x+4 \right) }{ 1 \cdot 120 } \xlongequal{\text{Step 3}} \frac{ x^5+4x^4+6x^4+24x^3+11x^3+44x^2+6x^2+24x }{ 120 } = \\[1ex] &= \frac{x^5+10x^4+35x^3+50x^2+24x}{120} \end{aligned} $$