Subtract: $x^{3} + 2 x^{2} y + 6 x y^{2} - y^{3}$ from $y^{3} - 3 x y^{2} - 4 x^{2} y$

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Solution

The subtraction is carried out as follows,
$(y^{3} - 3 x y^{2} - 4 x^{2} y) - (x^{3} + 2 x^{2} y + 6 x y^{2} - y^{3})$
$= y^{3} - 3 x y^{2} - 4 x^{2} y - x^{3} - 2 x^{2} y - 6 x y^{2} + y^{3}$
Rearranging the like terms together,
$= (y^{3} + y^{3}) + (- 3 x y^{2} - 6 x y^{2}) + (- 4 x^{2} y - 2 x^{2} y) - x^{3}$
$= 2 y^{3} - 9 x y^{2} - 6 x^{2} y - x^{3}$

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