To simplify the given expression, we first factor the denominators of each fraction:
x^2 - 6x - 27 factors to (x-9)(x+3)
x^2 - 6x + 9 factors to (x-3)(x-3)
x^2 - x - 20 factors to (x-5)(x+4)
x^2 + 10x + 24 factors to (x+6)(x+4)
Now, the expression becomes:
[(x-9)(x+3)/(x-3)(x-3)] - [(x-5)(x+4)/(x+6)(x+4)]
Now, we simplify each fraction:
(x-9)(x+3)/(x-3)(x-3) = (x-9)/(x-3)
(x-5)(x+4)/(x+6)(x+4) = (x-5)/(x+6)
Therefore, the final simplified expression is:
(x-9)/(x-3) - (x-5)/(x+6)
(x^(2)-6x-27)/(x^(2)-6x+9)-(x^(2)-x-20)/(x^(2)+10x+24)
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