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I'm certain that the question is not correctly asked. Notice that in the sentence,

How can I calculate this sum?


there is no sum at all!

However, it seems that the OP wants to know if the expresion $$\frac{1}{1 - (x + x^2)^2} = \sum_{n = 0}^\infty (x + x^2)^{2n},$$ is valid.

If that is the question... I'd say yes, as long as $(x + x^2)^2 <1.$