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:

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

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

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

Combine like terms:

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

Multiply each term of $ \left( \color{blue}{x^3+3x^2-6x-8}\right) $ by each term in $ \left( 3x+7\right) $.

$$ \left( \color{blue}{x^3+3x^2-6x-8}\right) \cdot \left( 3x+7\right) = 3x^4+7x^3+9x^3+21x^2-18x^2-42x-24x-56 $$

Combine like terms:

$$ 3x^4+ \color{blue}{7x^3} + \color{blue}{9x^3} + \color{red}{21x^2} \color{red}{-18x^2} \color{green}{-42x} \color{green}{-24x} -56 = 3x^4+ \color{blue}{16x^3} + \color{red}{3x^2} \color{green}{-66x} -56 $$