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:

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

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

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

Combine like terms:

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

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

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

Combine like terms:

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

Multiply each term of $ \left( \color{blue}{x^4+10x^3+35x^2+50x+24}\right) $ by each term in $ \left( x+5\right) $.

$$ \left( \color{blue}{x^4+10x^3+35x^2+50x+24}\right) \cdot \left( x+5\right) = \\ = x^5+5x^4+10x^4+50x^3+35x^3+175x^2+50x^2+250x+24x+120 $$

Combine like terms:

$$ x^5+ \color{blue}{5x^4} + \color{blue}{10x^4} + \color{red}{50x^3} + \color{red}{35x^3} + \color{green}{175x^2} + \color{green}{50x^2} + \color{orange}{250x} + \color{orange}{24x} +120 = \\ = x^5+ \color{blue}{15x^4} + \color{red}{85x^3} + \color{green}{225x^2} + \color{orange}{274x} +120 $$