<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] Use $(a+b)+(a-b)=2a$
and $(a+b)^2+(a-b)^2=2a^2+2b^2$
and $(a-b)(a+b)=a^2-b^2$
Then
$$
\underbrace{(\color{#C00}{(x+2y)}+\color{#090}{(x-2y)})}_{2x}\left(\!\vphantom{2^2}\right.\underbrace{\color{#C00}{(x+2y)}^2+\color{#090}{(x-2y)}^2}_{2x^2+8y^2}-\underbrace{\color{#C00}{(x+2y)}\color{#090}{(x-2y)}}_{x^2-4y^2}\left.\!\vphantom{2^2}\right)
$$
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