(12-x-x^2)/(x^2-4+3)
Can someone help me with this?
It's not magic. You have to look at it and encourage it to make sense.
The numerator is a little funny, so switch it around so that it looks like you expect:
12-x-x^2 = -x^2 - x + 12 = -(x^2 + x - 12)
That's better. Now it is easy to see that we need 4*(-3) = 12 and 4 + (-3) = +1
The denominator should be an eyeball problem.
(-1)*(-3) = +3 and (-1)+(-3) = -4
Once you do what TK said, look for binomials that are (a + b) to cancel.
like this: $\displaystyle \frac{\rlap{---------}(x + 5)(x - 2)}{\rlap{---------}(x + 5)} = x - 2$
but this doesn't cancel: $\displaystyle \frac{4 - {\color {red}{\not x}}}{{\color {red} {\not x}}}$