<s>[INST] Simplifying rational fractions I can't get this one either for whatever reason, spent about 20 minutes on it and I have made no progress at all.
$$\frac{x^2}{(x^2-4)} - \frac{x+1}{x+2}.$$
I know that I can simplify this into one fraction so I make it $$\frac{x^2}{x+2}-\frac{(x+1)(x^2-4)}{(x^2-4)(x+2)}$$
I then can simplify it further making the $(x^2+4)$ into $(x-2)(x+2)$ and the $(x+2)$ into $(x-1)(x+1)$ but this does not help me get the answer. I know I have to manipulate it in some counter intuitive way but I can not make it work.
 [/INST] $$\frac{x^2}{x^2-4} - \frac{x+1}{x+2} = \frac{x^2}{(x-2)(x+2)}-\frac{x+1}{x+2}=\frac{x^2}{(x-2)(x+2)}-\frac{(x-2)(x+1)}{(x-2)(x+2)}.$$
Working with the numerators:
$$
x^2-(x-2)(x+1) = x^2 - (x^2 -x -2).
$$
Here's the easiest mistake to make (I've seen this happen zillions of times including in calculus courses):
Right: $x^2 - (x^2 -x -2) = x^2 - x^2 + x + 2$
Wrong: $x^2 - (x^2 -x -2) = x^2 - x^2 - x - 2$
The numerator ends up being $x+2$, so we get another simplification:
$$
\frac{x+2}{(x-2)(x+2)} = \frac{1}{x-2}.
$$
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