Cauchy product of power series with $x^{2n}$

Question

I am trying to rewrite a function $y(x)=\frac{1}{1+x+x^2+x^3}$ as a series. I used geometric series and got to two power series that I need to Cauchy product, however, one of them has $x^{2n}$ and I cannot find anywhere how to do it. Here is the the product that I am trying to solve $$(\sum_{n=0}^\infty{(-1)^n x^n})(\sum_{n=0}^\infty{(-1)^n x^{2n}})$$