Find $ \left(x+2y\right)^3 $ using formula

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

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

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

Find $ \left(x-2y\right)^3 $ using formula

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

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

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

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

$$ \left( \color{blue}{x^3+6x^2y+12xy^2+8y^3}\right) \cdot \left( x^3-6x^2y+12xy^2-8y^3\right) = \\ = x^6 -\cancel{6x^5y}+12x^4y^2 -\cancel{8x^3y^3}+ \cancel{6x^5y}-36x^4y^2+ \cancel{72x^3y^3}-48x^2y^4+12x^4y^2 -\cancel{72x^3y^3}+144x^2y^4 -\cancel{96xy^5}+ \cancel{8x^3y^3}-48x^2y^4+ \cancel{96xy^5}-64y^6 $$

Combine like terms:

$$ x^6 \, \color{blue}{ -\cancel{6x^5y}} \,+ \color{green}{12x^4y^2} \, \color{orange}{ -\cancel{8x^3y^3}} \,+ \, \color{blue}{ \cancel{6x^5y}} \, \color{red}{-36x^4y^2} + \, \color{green}{ \cancel{72x^3y^3}} \, \color{blue}{-48x^2y^4} + \color{red}{12x^4y^2} \, \color{red}{ -\cancel{72x^3y^3}} \,+ \color{green}{144x^2y^4} \, \color{orange}{ -\cancel{96xy^5}} \,+ \, \color{red}{ \cancel{8x^3y^3}} \, \color{green}{-48x^2y^4} + \, \color{orange}{ \cancel{96xy^5}} \,-64y^6 = x^6 \color{red}{-12x^4y^2} + \color{green}{48x^2y^4} -64y^6 $$