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

Find $ \left(x-y\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}{ y }$.

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

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

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

Combine like terms:

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

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

$$ \left( \color{blue}{x+4y}\right) \cdot \left( x^4-4x^3y+6x^2y^2-4xy^3+y^4\right) = \\ = x^5 -\cancel{4x^4y}+6x^3y^2-4x^2y^3+xy^4+ \cancel{4x^4y}-16x^3y^2+24x^2y^3-16xy^4+4y^5 $$

Combine like terms:

$$ x^5 \, \color{blue}{ -\cancel{4x^4y}} \,+ \color{green}{6x^3y^2} \color{orange}{-4x^2y^3} + \color{blue}{xy^4} + \, \color{blue}{ \cancel{4x^4y}} \, \color{green}{-16x^3y^2} + \color{orange}{24x^2y^3} \color{blue}{-16xy^4} +4y^5 = \\ = x^5 \color{green}{-10x^3y^2} + \color{orange}{20x^2y^3} \color{blue}{-15xy^4} +4y^5 $$