Multiply each term of $ \left( \color{blue}{x-3}\right) $ by each term in $ \left( x-1\right) $.

$$ \left( \color{blue}{x-3}\right) \cdot \left( x-1\right) = x^2-x-3x+3 $$

Combine like terms:

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

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

$$ \left( \color{blue}{x^2-4x+3}\right) \cdot \left( x+2\right) = x^3+2x^2-4x^2-8x+3x+6 $$

Combine like terms:

$$ x^3+ \color{blue}{2x^2} \color{blue}{-4x^2} \color{red}{-8x} + \color{red}{3x} +6 = x^3 \color{blue}{-2x^2} \color{red}{-5x} +6 $$

Multiply each term of $ \left( \color{blue}{x^3-2x^2-5x+6}\right) $ by each term in $ \left( x+2\right) $.

$$ \left( \color{blue}{x^3-2x^2-5x+6}\right) \cdot \left( x+2\right) = \\ = x^4+ \cancel{2x^3} -\cancel{2x^3}-4x^2-5x^2-10x+6x+12 $$

Combine like terms:

$$ x^4+ \, \color{blue}{ \cancel{2x^3}} \, \, \color{blue}{ -\cancel{2x^3}} \, \color{green}{-4x^2} \color{green}{-5x^2} \color{orange}{-10x} + \color{orange}{6x} +12 = x^4 \color{green}{-9x^2} \color{orange}{-4x} +12 $$