the least common denominator would be 1/(x+3)(x-1) not 1/(x+3)(x-1)^2 which is what you needA/(x+3)+B/(x-1)+C/(x-1).
Think about it the least common denominator of 1/2 + 1/2^2 +1/3
is 1/(2^2 *3)
going the other way if you wanted to break 1/(2^2 *3) down into a the most general sum of fractions you would have A/2 + B/2^2 +C/3
same is true of 1/(x-1)^2(x+3)