Find all polynomials
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
with real coefficients such that for all reals
![a,b,c](/media/m/3/6/4/36454fdb50fc50f021324b33a6b513e3.png)
such that
![ab+bc+ca = 0](/media/m/1/7/2/172974b25c958710fcce8099b4ae4c13.png)
we have the following relations
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Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab+bc+ca = 0$ we have the following relations
$$f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c).$$