Simplify the following algebraic fraction; (3x^2 - x - 2) / ((1/2)x + (1/3)).

First we need to factorise the numerator into two expressions. We can see one expression must start (3x + ?) and the other therefore must hold (x + ?), we know this because the two brackets must multiply together to generate 3x². Now we need to consider two numbers that will multiply together to give -2, this can be either +1 and -2 or -1 and +2. To gain the required -x in the original expression we see our factorisation must read: (3x +2)(x-1).Now we want to remove the fractional coefficients in the demonimator. We can do this by multiplying the top and bottom by 2x3=6 to get: (6(3x+2)(x-1))/(3x+2).The final step is to cancel terms in the demoninator and numerator that are equal. Cancelling (3x+2) leaves us with the simplified expression; 6(x-1).