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

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

Combine like terms:

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

Find $ \left(x^3-x\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^3 } $ and $ B = \color{red}{ x }$.

$$ \begin{aligned}\left(x^3-x\right)^2 = \color{blue}{\left( x^3 \right)^2} -2 \cdot x^3 \cdot x + \color{red}{x^2} = x^6-2x^4+x^2\end{aligned} $$

Remove the parentheses by changing the sign of each term within them.

$$ - \left( x^6-2x^4+x^2 \right) = -x^6+2x^4-x^2 $$

Combine like terms:

$$ \, \color{blue}{ \cancel{x^6}} \, \, \color{green}{ -\cancel{2x^4}} \,+2x^3+ \, \color{blue}{ \cancel{x^2}} \,-2x+1 \, \color{blue}{ -\cancel{x^6}} \,+ \, \color{green}{ \cancel{2x^4}} \, \, \color{blue}{ -\cancel{x^2}} \, = 2x^3-2x+1 $$