$$ (x-2)^4 = (x-2)^2 \cdot (x-2)^2 $$

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 }$.

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

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

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

Combine like terms:

$$ x^4 \color{blue}{-4x^3} + \color{red}{4x^2} \color{blue}{-4x^3} + \color{green}{16x^2} \color{orange}{-16x} + \color{green}{4x^2} \color{orange}{-16x} +16 = \\ = x^4 \color{blue}{-8x^3} + \color{green}{24x^2} \color{orange}{-32x} +16 $$

Find $ \left(x-1\right)^3 $ using formula

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

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

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

Combine like terms:

$$ x^4 \color{blue}{-8x^3} + \color{red}{24x^2} \color{green}{-32x} + \color{orange}{16} + \color{blue}{x^3} \color{red}{-3x^2} + \color{green}{3x} \color{orange}{-1} = \\ = x^4 \color{blue}{-7x^3} + \color{red}{21x^2} \color{green}{-29x} + \color{orange}{15} $$