<s>[INST] simplifying complex fractions I have this expression
${2 \over x^2 - 4}$ + ${1 \over x + 2}$
So I would take the LCM to ${x^2 - 4}$ or using the difference of 2 squares to be ${(x + 2)(x - 2)}$
So I would simplify this to:
${2 + (x - 2) \over (x - 2)(x + 2)}$ which would simplify further to
${2\over x + 2}$
But the answer to the question is ${x \over x^2 - 4}$
I'm not sure how the real answer is achieved and how I am wrong
 [/INST] Notice, we have $$\frac{2}{x^2-4}+\frac{1}{x+2}$$ $\color{red}{\text{apply} \ a^2-b^2=(a-b)(a+b)}$ $$=\frac{2}{(x-2)(x+2)}+\frac{1}{x+2}$$ $\color{red}{\text{take L.C.M of denominators }}$  $$=\frac{2+x-2}{(x-2)(x+2)}$$$$ =\frac{x}{(x-2)(x+2)}$$ Thus we have following simplified form  $$\bbox[5px, border: 2px solid #C0A000]{\color{red}{\frac{2}{x^2-4}+\frac{1}{x+2}=\frac{x}{(x-2)(x+2)}}}$$
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