<s>[INST] Can $A^2+B^2+C^2-2AB-2AC-2BC$ be a perfect square? I have come upon the trivariate polynomial $A^2+B^2+C^2-2AB-2AC-2BC$ and want to factor it.  Because of the symmetry, I am wondering if it can be a perfect square or if there is some other nice factorization.
 [/INST] A more symmetric factorization would be:
$$A^2 + B^2 + C^2 - 2AB -2AC - 2BC =$$
$$(A + B + C)^2 - 4AB - 4AC - 4BC =$$
$$(A + B + C - 2 \sqrt{AB + AC + BC}) (A + B + C + 2 \sqrt{AB + AC + BC})$$
if
$$AB + AC + BC \ge 0$$
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