<s>[INST] Decomposition of a sum of cubes I have to decompose the following polynomial:
$$(x+2y)^3+(x-2y)^3$$
Since I have the sum of two cubes, I have thought:
$$(x+2y)^3+(x-2y)^3=(x+2y+x-2y)[(x+2y)^2+(x-2y)^2-(x+2y)(x-2y)]=2x\color{red}{[(x+2y)^2+(x-2y)^2-(x+2y)(x-2y)]}$$
Now in my book the result is $2x(x^2+12y^2)$, but I can't understand how rewriting the red term as $x^2+12y^2$, without trivially executing  the squares. Can you give me a hint?
 [/INST] Compare $S = (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ 
with $T = (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3.$
Adding: $S + T = 2(a^3 + 3ab^2).$
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