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

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-1\right)^3 $ using formula

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

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

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

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 $ \color{blue}{2} $ by $ \left( x^4-4x^3+6x^2-4x+1\right) $ $$ \color{blue}{2} \cdot \left( x^4-4x^3+6x^2-4x+1\right) = 2x^4-8x^3+12x^2-8x+2 $$Multiply $ \color{blue}{16} $ by $ \left( x^3-3x^2+3x-1\right) $ $$ \color{blue}{16} \cdot \left( x^3-3x^2+3x-1\right) = 16x^3-48x^2+48x-16 $$Multiply $ \color{blue}{40} $ by $ \left( x^2-2x+1\right) $ $$ \color{blue}{40} \cdot \left( x^2-2x+1\right) = 40x^2-80x+40 $$Multiply $ \color{blue}{32} $ by $ \left( x-1\right) $ $$ \color{blue}{32} \cdot \left( x-1\right) = 32x-32 $$

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

$$ - \left( 16x^3-48x^2+48x-16 \right) = -16x^3+48x^2-48x+16 $$

Combine like terms:

$$ 2x^4 \color{blue}{-8x^3} + \color{red}{12x^2} \color{green}{-8x} + \color{orange}{2} \color{blue}{-16x^3} + \color{red}{48x^2} \color{green}{-48x} + \color{orange}{16} = \\ = 2x^4 \color{blue}{-24x^3} + \color{red}{60x^2} \color{green}{-56x} + \color{orange}{18} $$

Combine like terms:

$$ 2x^4-24x^3+ \color{blue}{60x^2} \color{red}{-56x} + \color{green}{18} + \color{blue}{40x^2} \color{red}{-80x} + \color{green}{40} = \\ = 2x^4-24x^3+ \color{blue}{100x^2} \color{red}{-136x} + \color{green}{58} $$

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

$$ - \left( 32x-32 \right) = -32x+32 $$

Combine like terms:

$$ 2x^4-24x^3+100x^2 \color{blue}{-136x} + \color{red}{58} \color{blue}{-32x} + \color{green}{32} + \color{green}{3} = 2x^4-24x^3+100x^2 \color{blue}{-168x} + \color{green}{93} $$