If a and b are coprime then a+b is also coprime to a and b.

Then any factor of a+b is coprime to a and b.

Let f be some common factor of a+b and a^2+b^2.

Then f is also a factor of (a+b)^2.

Then f is also a factor of (a+b)^2 - a^2+b^2 = 2ab.

Then f is a common factor of a+b and 2ab.

Since any factor of a+b is coprime to a and b, then f is a factor of 2.

So f is at most 2. QED.