Actually you don't have to expand the whole equation to see why A and C must be 0. Just imagine that you have multiplied both side already by the two denominators. Then the left-hand side would just be 3.
Now looking at the right-hand side, if A and C were non-zero, then the right-hand side would contain terms of the third degree (because a first-degree term of the numerator multiplied by the second-degree term of the denominator equals a third-degree term). Namely the coefficients of the third degree would be A+C. Since the left-hand side does not have any third degree terms, A+C = 0.
Therefore, we know that A and C are of equal magnitude but with opposite signs. Now look at the first-degree term on the right-hand side and we know that 9A+4C=0, which means A and C can't actually be of equal magnitude if they were non-zero. Therefore they are both zero.
So your textbook could simply have said examine the odd powers of x on the right-hand side of the equation to anticipate that A=C=0.