Simplify (3x^2-x-2)/(x^2-1)

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This is an algebraic fraction. There is more than one way of solving this expression, but the simplest is to factorise both the top and the bottom quadratic expressions.

Firstly we shall simplify the top expression, 3x^2-x-2. This means putting it in the form (ax + b)(cx + d) where a, b, c, and d are constants. 

Now, the only way to get 3x^2, you can see that a multipled by c must equal to 3. Therefore, (3x + b)(x + d). Now you can also see that b multiplied by d must be -2, but also, 3 multipled by d plus b multiplied by x must also be -1. Therefore it can be seen that b is 2, and d is -1. So now the top is (3x + 2)(x -1).

Next, we factorise the bottom of the fraction. At first glance, it looks like it is the simplest it can be. However, x^2 - 1 is actually the difference of two squares. This is (x + 1)(x - 1).

Now, the full factorised expression becomes (3x + 2)(x - 1)/(x + 1)(x - 1). As (x - 1) is on the top and the bottom, this is equal to 1, and can be cancelled out.

So now we have the answer! (3x + 2)/(x + 1)

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