$$ (k-1)^4 = (k-1)^2 \cdot (k-1)^2 $$

Find $ \left(k-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}{ k } $ and $ B = \color{red}{ 1 }$.

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

Multiply each term of $ \left( \color{blue}{k^2-2k+1}\right) $ by each term in $ \left( k^2-2k+1\right) $.

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

Combine like terms:

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

Multiply $ \color{blue}{k} $ by $ \left( k^4-4k^3+6k^2-4k+1\right) $ $$ \color{blue}{k} \cdot \left( k^4-4k^3+6k^2-4k+1\right) = k^5-4k^4+6k^3-4k^2+k $$

Multiply each term of $ \left( \color{blue}{k^5-4k^4+6k^3-4k^2+k}\right) $ by each term in $ \left( k-2\right) $.

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

Combine like terms:

$$ k^6 \color{blue}{-2k^5} \color{blue}{-4k^5} + \color{red}{8k^4} + \color{red}{6k^4} \color{green}{-12k^3} \color{green}{-4k^3} + \color{orange}{8k^2} + \color{orange}{k^2} -2k = \\ = k^6 \color{blue}{-6k^5} + \color{red}{14k^4} \color{green}{-16k^3} + \color{orange}{9k^2} -2k $$