The coefficient of x4 in the expansion of 1+x2−x−1 in ascending powers of x, when |x|<1, is

1+x2−x−1=11+x2−x×1+x2+x1+x2+x=1+x2+x1+x2−x2=x+1+x2=x+1+x21/2=x+1+12x2+12−12x42!+⋯Therefore, the coefficient of x4 is -1/8.