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

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

Combine like terms:

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

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

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

Combine like terms:

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

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

$$ \left( \color{blue}{x^3+x^2-4x-4}\right) \cdot \left( x+3\right) = x^4+3x^3+x^3+3x^2-4x^2-12x-4x-12 $$

Combine like terms:

$$ x^4+ \color{blue}{3x^3} + \color{blue}{x^3} + \color{red}{3x^2} \color{red}{-4x^2} \color{green}{-12x} \color{green}{-4x} -12 = x^4+ \color{blue}{4x^3} \color{red}{-x^2} \color{green}{-16x} -12 $$

$$ \left( \color{blue}{x^4+4x^3-x^2-16x-12}\right) \cdot x = x^5+4x^4-x^3-16x^2-12x $$