Can someone help me with this complex fraction question?

let's see what I get:

[(x - 2) / (x + 2) + (x - 1) / (x + 1)] / [x / (x + 1) - (2x - 3) / x]

IF that's what the problem is, then let's start with getting common denominators. In the numerator we get (x + 2)(x + 1) and in the denominator we get x(x + 1). So now we have:

{(x - 2)(x + 1) / [(x + 1)(x + 2)] + (x - 1)(x + 2) / [(x - 2)(x + 1)]} / {x² / [x(x + 1)] - (2x - 3)(x + 1) / [x(x + 1)]}

Now add or subtract your numerators:

{[(x - 2)(x + 1) + (x - 1)(x + 2)] / [(x + 2)(x + 1)]} / {[x² - (2x - 3)(x + 1)] / [x(x + 1)]}

Now let's simplify what we have before we get more complicated, then re-factor what you can:

(x² - x - 2 + x² + x - 2) / [(x + 2)(x + 1)]} / {[x² - (2x² + 2x - 3x - 3)] / [x(x + 1)]}

(2x² - 4) / [(x + 2)(x + 1)]} / {[x² - (2x² - x - 3)] / [x(x + 1)]}

2(x² - 2) / [(x + 2)(x + 1)]} / {(x² - 2x² + x + 3) / [x(x + 1)]}

2(x² - 2) / [(x + 2)(x + 1)]} / {(-x² + x + 3) / [x(x + 1)]}

Multiplying both halves by -1 to get the negative out of the high-order coefficient:

-2(x² - 2) / [(x + 2)(x + 1)]} / {(x² - x - 3) / [x(x + 1)]}

Now we can turn the division of fractions into the multiplication of the reciprocal:

-2(x² - 2) / [(x + 2)(x + 1)]} * [x(x + 1) / (x² - x - 3)]

(x + 1) cancels out from the first denominator and the second numerator:

-2(x² - 2) / (x + 2) * x / (x² - x - 3)

Multiply your numerators and denominators:

-2x(x² - 2) / [(x + 2)(x² - x - 3)]

Now expand the halves and that is the simplified form:

(-2x³ + 4x) / (x³ - x² - 3x + 2x² - 2x - 6)

(-2x³ + 4x) / (x³ + x² - 5x - 6)

So, I don't get what you get, and I don't get what the answer is supposed to be, but I'm getting something in the middle. Hope you can follow my steps and see what went wrong.

[((x - 2)/(x + 2)) + ((x - 1)/(x + 1))] / [((x/(x + 1)) - ((2x - 3)/x))]

= [2(x^2 - 2)/(x^2 + 3x + 2)] / [(-x^2 + x + 3)/(x^2 + x)]

= [2(x^2 - 2)(x^2 + x)] / [(x^2 + 3x + 2)(-x^2 + x + 3)]

= [-2x(x^2 - 2)] / [(x + 2)(x^2 - x - 3)]