Answer
$$x^3y+y^3(-x-y)+x(-x-y)^3=x^3y-xy^3-y^4-x^4-3x^3y-3x^2y^2-xy^3$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}x^3y+y^3(-x-y)+x(-x-y)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}x^3y+y^3(-x-y)+x(-x^3-3x^2y-3xy^2-y^3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}x^3y-xy^3-y^4-x^4-3x^3y-3x^2y^2-xy^3\end{aligned} $$ | |
| ① | Find $ \left(-x-y\right)^3 $ using formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = -x $ and $ B = y $. $$ \left(-x-y\right)^3 = \left( -x \right)^3-3 \cdot \left( -x \right)^2 \cdot y + 3 \cdot \left( -x \right) \cdot y^2-y^3 = -x^3-3x^2y-3xy^2-y^3 $$ |
| ② | Combine like terms: $$ x^3y-xy^3-y^4 = x^3y-xy^3-y^4 $$ |
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