<s>[INST] How to factorize $2ab^2+2b^2c-ac^2-bc^2-a^2b-a^2c$? If $a,b,c$ are in AP $\implies \frac{1}{bc}, \frac{1}{ca}, \frac{1}{ab}$ are in AP $\implies
\frac{ab+bc+ca}{bc}, \frac{ab+bc+ca}{ca}, \frac{ab+bc+ca}{ab}$ are in AP $\implies \frac{bc+ca}{bc},\frac{ab+bc}{ca},\frac{bc+ca}{ab}$ are in AP $\implies \frac{bc}{bc+ca}, \frac{ca}{ab+bc}, \frac{ab}{bc+ca}$
are in HP.
If $P,Q,R$ are in HP  $\implies Q=\frac{2PR}{P+R}.$ By this method proving that  $\frac{bc}{bc+ca}, \frac{ca}{ab+bc}, \frac{ab}{bc+ca}$, if $a,b,c$ are in AP; is not straight forward.
Curiously, then we need to factorize  $(2ab^2+2b^2c-ac^2-bc^2-a^2b-a^2c)$.
The question is: How do you factorize the last expression?
 [/INST] Given expression is $2(a+c)b^2-(a^2+c^2)b-ac(a+c)$
=$2(a+c)b^2-(a+c)^2b+2acb - ac(a+c)=(a+c)b\left[2b-(a+c) \right]+ac\left[2b-(a+c)\right]$
$=(ab+bc+ca)(2a-b-c)$
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