If this had factored, both factors would be linear, so the graph of the following equation would have been 1 or 2 planes
However, the graph is a line x=y=z as can be seen by using the Cauchy-Schwarz inequality:
(a12 + ... + an2)(b12 + ... + bn2) ≥ (a1b1 + ... + anbn)2
With equality happening if for all 1 ≤ k ≤ n, ak = bk.
In this case, take (a1,a2,a3) = (x,y,z) and (b1,b2,b3) = (y,z,x).
(x2 + y2 + z2)(y2 + z2 + x2) ≥ (xy + yz + zx)2
|x2 + y2 + z2| ≥ |xy + yz + zx|
with equality only along the line x=y=z.
The conclusion, is that it doesn't factor in the ordinary sense.