2x^2+x-6 factorises: (x+2)(2x-3) so we have (x+2)(2x-3)/(x+2). There is a common factor in the denominator and numerator: x+2, so divide them by this common factor: 2x-3 is the answer. There's just one thing: you can only divide through by a common factor if the factor isn't zero, in other words x+2 cannot be zero, so x cannot be -2. Any other value except x=-2 is OK. This is because we can't divide by zero.
How did I know how to factorise the top? Well, I guessed that the denominator was likely to be a factor, so I used the "solution" of x+2=0, subtracting 2 from each side: x=-2, then I substituted this value of x into 2x^2+x-6: 2*(-2)*(-2)-2-6=8-2-6=0, which tells me that x+2 divides into the numerator. To find the other factor making up the quadratic, I asked myself what do I need to multiply x by to get 2x^2, and the answer is 2x^2/x=2x; then I asked, what do I need to multiply 2 by to get 6? 6/2=3. Finally, what sign + or - is needed to go in front of 3? We already have +2 in the factor x+2, so we need a minus to make -6. So the other factor of the quadratic must be 2x-3.