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

First notice that no simplifications can be made to the problem without changing the form. We start by factorising the numerator. The highest power of x in this question is 2, therefore we know that it has to be a quadratic. The general form of a quadratic is Ax²+Bx+C. We need to find 2 factors of -2 that add up to 1 when one of the factors is multiplied by 3 (as A = 3). The possible factors are -1 and 2 or -2 and 1. Start with -1 and 2. The two possibilities are: 1) (3x - 1)(x + 2)- the sum of the coefficients of x is 5. 2) (3x + 2)(x - 1) - the sum of the coefficients of x is -1Then try -2 and 1. The two possibilities are: 1) (3x + 1)(x - 2) - the sum of the coefficients of x is -5. 2) (3x - 2)(x + 1) - the sum of the coefficients of x is 1.Therefore the factorised form is (3x - 2)(x + 1).The denominator can then be simplified very easily by noticing that x² -1 is in the form a²-b². We know that a²-b² = (a+b)(a-b). Applying this to the denominator gives (x+1)(x-1). To simplify (3x² + x -2)/(x² - 1) we rewrite it in the form [(3x - 2)(x + 1)/(x + 1)(x - 1)] and cancel x + 1 giving the simplified form as (3x - 2)/(x - 1).