<s>[INST] Question from Spivak's Calculus. Hint makes no sense. I'm currently working through Spivak. This question has me a little bit tied up in knots. I was able to answer it, but I can't make heads or tails of the hint.
Here's the question:
Find out when $(x+y)^5=x^5+y^5$. Hint: from the assumption $(x+y)^5=x^5+y^5$ you should be able to derive the equation $x^3 + 2x^2y + 2xy^2+y^3= 0$ if $xy\neq 0$. (I got this part, the next part is what confuses me.) This implies that $(x+y)^3=x^2y + xy^2=xy(x+y)$. 
The last part of the above is what has me confused. What he's stating to be equal to $(x+y)^3$looks to me to be the difference of the binomial expansion of $(x+y)^3$ and the other equation given in the hint. Am I missing something???
 [/INST] $$(x+y)^3 = x^3+3x^2y+3xy^2+y^3.$$
Suppose $x^3 + 2x^2y+2xy^2+y^3 = 0$. Add to both sides $x^2y+xy^2$:
$$x^3+2x^2y+2xy^2+y^3 + \color{red}{x^2y+xy^2} = \color{red}{x^2y+xy^2} \\
x^3+3x^2y+3xy^2+y^3 = x^2y+xy^2 \\
(x+y)^3 = xy(x+y).$$
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