\frac{x^2-x^{2n+4}}{1-x^2} = \frac{x^2-x^{2n+2}}{1-x^2} + x^{2n+4} = \frac{x^2-x^{2n+2}+x^{2n+2}(1-x^2)}{1-x^2} = \frac{x^2-x^{2n+2}+x^{2n+2}-x^{2n+4}}{1-x^2} = \frac{x^2-x^{2n+4}}{1-x^2}