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

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+2\right)^3 $ using formula

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

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

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

Multiply each term of $ \left( \color{blue}{x^4-8x^3+24x^2-32x+16}\right) $ by each term in $ \left( x^3+6x^2+12x+8\right) $.

$$ \left( \color{blue}{x^4-8x^3+24x^2-32x+16}\right) \cdot \left( x^3+6x^2+12x+8\right) = \\ = x^7+6x^6+12x^5+8x^4-8x^6-48x^5-96x^4-64x^3+24x^5+144x^4+288x^3+192x^2-32x^4-192x^3-384x^2-256x+16x^3+96x^2+192x+128 $$

Combine like terms:

$$ x^7+ \color{blue}{6x^6} + \color{red}{12x^5} + \color{green}{8x^4} \color{blue}{-8x^6} \color{orange}{-48x^5} \color{blue}{-96x^4} \color{red}{-64x^3} + \color{orange}{24x^5} + \color{green}{144x^4} + \color{orange}{288x^3} + \color{blue}{192x^2} \color{green}{-32x^4} \color{red}{-192x^3} \color{green}{-384x^2} \color{orange}{-256x} + \color{red}{16x^3} + \color{green}{96x^2} + \color{orange}{192x} +128 = \\ = x^7 \color{blue}{-2x^6} \color{orange}{-12x^5} + \color{green}{24x^4} + \color{red}{48x^3} \color{green}{-96x^2} \color{orange}{-64x} +128 $$