A GCD question

I know it's easy to find the GCD of two integers using Euclid's
algorithm;
it's not much more difficult to find the GCD if two polynomials in a
similiar way.

But is it possible to do this for a multinomial?
I've been trying and failing to simplify
GCD(x + y, x^2 -x*y + y^2) - is this doomed to fail?

Regards,
Jeremy

.

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