Final answer:
The simplified expression of (x+1)/(x+3)-(x+1)/(3-x)-(2x(x-1))/((x²)-9) is -2x² + 8x - 6.
Step-by-step explanation:
To simplify the expression (x+1)/(x+3)-(x+1)/(3-x)-(2x(x-1))/((x²)-9), we can follow these steps:
Step 1: Simplify the denominators and combine the fractions if possible.
- In the first fraction, (x+1)/(x+3), the denominator is (x+3).
- In the second fraction, (x+1)/(3-x), the denominator is (3-x).
The denominators are different, so we need to find a common denominator.
The common denominator can be obtained by multiplying the two denominators together:
(x+3)(3-x) = (x+3)(-1)(x-3) = -(x+3)(x-3) = -(x²-9) = -x²+9
Now, let's rewrite the fractions with the common denominator:
(x+1)/(x+3) = [(x+1)(-1)(x-3)] / [-x²+9] = (-x²+4x-3) / (-x²+9)
(x+1)/(3-x) = [(x+1)(x+3)] / [-x²+9] = (x²+4x+3) / (-x²+9)
Step 2: Simplify the expression by subtracting the fractions.
(x+1)/(x+3) - (x+1)/(3-x) = [(-x²+4x-3) / (-x²+9)] - [(x²+4x+3) / (-x²+9)]
To subtract the fractions, we need a common denominator, which is already -x²+9.
Now, subtract the numerators:
(-x²+4x-3) - (x²+4x+3) = -x²+4x-3 - x²-4x-3
Combine like terms:
-2x² + 8x - 6
Finally, the simplified expression is -2x² + 8x - 6.
Your question is incomplete, but most probably the full question was:
Simplify the epression:
(x+1)/(x+3)-(x+1)/(3-x)-(2x(x-1))/((x²)-9)