The Answer is : (x - 3)/(x + 2) not x+2/x+3 Thus A) is your Answer
Simplify the following:
((x^2 + x - 6) (x^2 - 9))/((x^2 - 4) (x^2 + 6 x + 9))
The factors of -6 that sum to are 3 and -2. So, x^2 + x - 6 = (x + 3) (x - 2):
((x + 3) (x - 2) (x^2 - 9))/((x^2 - 4) (x^2 + 6 x + 9))
The factors of 9 that sum to 6 are 3 and 3. So, x^2 + 6 x + 9 = (x + 3) (x + 3):
((x + 3) (x - 2) (x^2 - 9))/((x + 3) (x + 3) (x^2 - 4))
(x + 3) (x + 3) = (x + 3)^2:
((x + 3) (x - 2) (x^2 - 9))/((x + 3)^2 (x^2 - 4))
x^2 - 4 = x^2 - 2^2:
((x + 3) (x - 2) (x^2 - 9))/((x^2 - 2^2) (x + 3)^2)
Factor the difference of two squares. x^2 - 2^2 = (x - 2) (x + 2):
((x + 3) (x - 2) (x^2 - 9))/((x - 2) (x + 2) (x + 3)^2)
x^2 - 9 = x^2 - 3^2:
((x + 3) (x - 2) (x^2 - 3^2))/((x - 2) (x + 2) (x + 3)^2)
Factor the difference of two squares. x^2 - 3^2 = (x - 3) (x + 3):
((x - 3) (x + 3) (x + 3) (x - 2))/((x - 2) (x + 2) (x + 3)^2)
((x + 3) (x - 2) (x - 3) (x + 3))/((x - 2) (x + 2) (x + 3)^2) = (x - 2)/(x - 2)×((x + 3) (x - 3) (x + 3))/((x + 2) (x + 3)^2) = ((x + 3) (x - 3) (x + 3))/((x + 2) (x + 3)^2):
((x + 3) (x - 3) (x + 3))/((x + 2) (x + 3)^2)
Combine powers. ((x + 3) (x - 3) (x + 3))/((x + 2) (x + 3)^2) = ((x + 3)^(1 + 1) (x - 3))/((x + 2) (x + 3)^2):
((x + 3)^(1 + 1) (x - 3))/((x + 3)^2 (x + 2))
1 + 1 = 2:
((x + 3)^2 (x - 3))/((x + 2) (x + 3)^2)
Cancel terms. ((x + 3)^2 (x - 3))/((x + 2) (x + 3)^2) = (x - 3)/(x + 2):
Answer: (x - 3)/(x + 2)