Multiply each term of $ \left( \color{blue}{x+5}\right) $ by each term in $ \left( x+1\right) $.

$$ \left( \color{blue}{x+5}\right) \cdot \left( x+1\right) = x^2+x+5x+5 $$

Combine like terms:

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

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

$$ \left( \color{blue}{x^2+6x+5}\right) \cdot \left( x+2\right) = x^3+2x^2+6x^2+12x+5x+10 $$

Combine like terms:

$$ x^3+ \color{blue}{2x^2} + \color{blue}{6x^2} + \color{red}{12x} + \color{red}{5x} +10 = x^3+ \color{blue}{8x^2} + \color{red}{17x} +10 $$

Multiply each term of $ \left( \color{blue}{x^3+8x^2+17x+10}\right) $ by each term in $ \left( x+3\right) $.

$$ \left( \color{blue}{x^3+8x^2+17x+10}\right) \cdot \left( x+3\right) = x^4+3x^3+8x^3+24x^2+17x^2+51x+10x+30 $$

Combine like terms:

$$ x^4+ \color{blue}{3x^3} + \color{blue}{8x^3} + \color{red}{24x^2} + \color{red}{17x^2} + \color{green}{51x} + \color{green}{10x} +30 = x^4+ \color{blue}{11x^3} + \color{red}{41x^2} + \color{green}{61x} +30 $$

Multiply $x^4+11x^3+41x^2+61x+30$ by $ \dfrac{x+4}{120} $ to get $ \dfrac{x^5+15x^4+85x^3+225x^2+274x+120}{120} $.

Step 1: Write $ x^4+11x^3+41x^2+61x+30 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} x^4+11x^3+41x^2+61x+30 \cdot \frac{x+4}{120} & \xlongequal{\text{Step 1}} \frac{x^4+11x^3+41x^2+61x+30}{\color{red}{1}} \cdot \frac{x+4}{120} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^4+11x^3+41x^2+61x+30 \right) \cdot \left( x+4 \right) }{ 1 \cdot 120 } \xlongequal{\text{Step 3}} \frac{ x^5+4x^4+11x^4+44x^3+41x^3+164x^2+61x^2+244x+30x+120 }{ 120 } = \\[1ex] &= \frac{x^5+15x^4+85x^3+225x^2+274x+120}{120} \end{aligned} $$