Find $ \left(x-2\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}{ 2 }$.

$$ \begin{aligned}\left(x-2\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 2 + \color{red}{2^2} = x^2-4x+4\end{aligned} $$$$ (x+1)^4 = (x+1)^2 \cdot (x+1)^2 $$

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^2+2x+1}\right) $ by each term in $ \left( x^2+2x+1\right) $.

$$ \left( \color{blue}{x^2+2x+1}\right) \cdot \left( x^2+2x+1\right) = x^4+2x^3+x^2+2x^3+4x^2+2x+x^2+2x+1 $$

Combine like terms:

$$ x^4+ \color{blue}{2x^3} + \color{red}{x^2} + \color{blue}{2x^3} + \color{green}{4x^2} + \color{orange}{2x} + \color{green}{x^2} + \color{orange}{2x} +1 = x^4+ \color{blue}{4x^3} + \color{green}{6x^2} + \color{orange}{4x} +1 $$

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)^3 $ using formula

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

where $ A = x $ and $ B = 1 $.

$$ \left(x+1\right)^3 = x^3+3 \cdot x^2 \cdot 1 + 3 \cdot x \cdot 1^2+1^3 = x^3+3x^2+3x+1 $$

Multiply $ \color{blue}{3} $ by $ \left( x^2-4x+4\right) $ $$ \color{blue}{3} \cdot \left( x^2-4x+4\right) = 3x^2-12x+12 $$$$ \left( \color{blue}{x^3-6x^2+12x-8}\right) \cdot 4 = 4x^3-24x^2+48x-32 $$

Multiply each term of $ \left( \color{blue}{3x^2-12x+12}\right) $ by each term in $ \left( x^4+4x^3+6x^2+4x+1\right) $.

$$ \left( \color{blue}{3x^2-12x+12}\right) \cdot \left( x^4+4x^3+6x^2+4x+1\right) = \\ = 3x^6+ \cancel{12x^5}+18x^4+12x^3+3x^2 -\cancel{12x^5}-48x^4-72x^3-48x^2-12x+12x^4+48x^3+72x^2+48x+12 $$

Combine like terms:

$$ 3x^6+ \, \color{blue}{ \cancel{12x^5}} \,+ \color{green}{18x^4} + \color{orange}{12x^3} + \color{blue}{3x^2} \, \color{blue}{ -\cancel{12x^5}} \, \color{red}{-48x^4} \color{green}{-72x^3} \color{orange}{-48x^2} \color{blue}{-12x} + \color{red}{12x^4} + \color{green}{48x^3} + \color{orange}{72x^2} + \color{blue}{48x} +12 = \\ = 3x^6 \color{red}{-18x^4} \color{green}{-12x^3} + \color{orange}{27x^2} + \color{blue}{36x} +12 $$

Multiply each term of $ \left( \color{blue}{4x^3-24x^2+48x-32}\right) $ by each term in $ \left( x^3+3x^2+3x+1\right) $.

$$ \left( \color{blue}{4x^3-24x^2+48x-32}\right) \cdot \left( x^3+3x^2+3x+1\right) = \\ = 4x^6+12x^5+12x^4+4x^3-24x^5-72x^4-72x^3-24x^2+48x^4+144x^3+144x^2+48x-32x^3-96x^2-96x-32 $$

Combine like terms:

$$ 4x^6+ \color{blue}{12x^5} + \color{red}{12x^4} + \color{green}{4x^3} \color{blue}{-24x^5} \color{orange}{-72x^4} \color{blue}{-72x^3} \color{red}{-24x^2} + \color{orange}{48x^4} + \color{green}{144x^3} + \color{orange}{144x^2} + \color{blue}{48x} \color{green}{-32x^3} \color{orange}{-96x^2} \color{blue}{-96x} -32 = \\ = 4x^6 \color{blue}{-12x^5} \color{orange}{-12x^4} + \color{green}{44x^3} + \color{orange}{24x^2} \color{blue}{-48x} -32 $$

Combine like terms:

$$ \color{blue}{3x^6} \color{red}{-18x^4} \color{green}{-12x^3} + \color{orange}{27x^2} + \color{blue}{36x} + \color{red}{12} + \color{blue}{4x^6} -12x^5 \color{red}{-12x^4} + \color{green}{44x^3} + \color{orange}{24x^2} \color{blue}{-48x} \color{red}{-32} = \\ = \color{blue}{7x^6} -12x^5 \color{red}{-30x^4} + \color{green}{32x^3} + \color{orange}{51x^2} \color{blue}{-12x} \color{red}{-20} $$