Find $ \left(x-2\right)^3 $ using formula

$$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$

where $ A = x $ and $ B = 2 $.

$$ \left(x-2\right)^3 = x^3-3 \cdot x^2 \cdot 2 + 3 \cdot x \cdot 2^2-2^3 = x^3-6x^2+12x-8 $$

Find $ \left(x+1\right)^2 $ using formula.

$$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$

where $ A = \color{blue}{ x } $ and $ B = \color{red}{ 1 }$.

$$ \begin{aligned}\left(x+1\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 1 + \color{red}{1^2} = x^2+2x+1\end{aligned} $$

Multiply each term of $ \left( \color{blue}{x^3-6x^2+12x-8}\right) $ by each term in $ \left( x^2+2x+1\right) $.

$$ \left( \color{blue}{x^3-6x^2+12x-8}\right) \cdot \left( x^2+2x+1\right) = \\ = x^5+2x^4+x^3-6x^4 -\cancel{12x^3}-6x^2+ \cancel{12x^3}+24x^2+12x-8x^2-16x-8 $$

Combine like terms:

$$ x^5+ \color{blue}{2x^4} + \color{red}{x^3} \color{blue}{-6x^4} \, \color{green}{ -\cancel{12x^3}} \, \color{blue}{-6x^2} + \, \color{green}{ \cancel{12x^3}} \,+ \color{red}{24x^2} + \color{green}{12x} \color{red}{-8x^2} \color{green}{-16x} -8 = \\ = x^5 \color{blue}{-4x^4} + \color{green}{x^3} + \color{red}{10x^2} \color{green}{-4x} -8 $$

Multiply each term of $ \left( \color{blue}{x^5-4x^4+x^3+10x^2-4x-8}\right) $ by each term in $ \left( x+3\right) $.

$$ \left( \color{blue}{x^5-4x^4+x^3+10x^2-4x-8}\right) \cdot \left( x+3\right) = \\ = x^6+3x^5-4x^5-12x^4+x^4+3x^3+10x^3+30x^2-4x^2-12x-8x-24 $$

Combine like terms:

$$ x^6+ \color{blue}{3x^5} \color{blue}{-4x^5} \color{red}{-12x^4} + \color{red}{x^4} + \color{green}{3x^3} + \color{green}{10x^3} + \color{orange}{30x^2} \color{orange}{-4x^2} \color{blue}{-12x} \color{blue}{-8x} -24 = \\ = x^6 \color{blue}{-x^5} \color{red}{-11x^4} + \color{green}{13x^3} + \color{orange}{26x^2} \color{blue}{-20x} -24 $$