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 each term of $ \left( \color{blue}{x^4+6x^3+11x^2+6x}\right) $ by each term in $ \left( x+4\right) $.

$$ \left( \color{blue}{x^4+6x^3+11x^2+6x}\right) \cdot \left( x+4\right) = x^5+4x^4+6x^4+24x^3+11x^3+44x^2+6x^2+24x $$

Combine like terms:

$$ x^5+ \color{blue}{4x^4} + \color{blue}{6x^4} + \color{red}{24x^3} + \color{red}{11x^3} + \color{green}{44x^2} + \color{green}{6x^2} +24x = \\ = x^5+ \color{blue}{10x^4} + \color{red}{35x^3} + \color{green}{50x^2} +24x $$