<s>[INST] When are both fractions integers? The sum of absolute values of all real numbers $x$, such that both of the fractions $\displaystyle \frac{x^2+4x−17}{x^2−6x−5}$ and $\displaystyle \frac{1−x}{1+x}$ are integers, can be written as $\displaystyle \frac{a}{b}$, where $a$ and $b$ are coprime positive integers. What is the value of $a+b$?
 [/INST] Hints:
$$\text{I}\;\;\;\;\;\;\;\frac{x^2+4x−17}{x^2−6x−5}=1+\frac{10x-12}{x^2-6x-5}=1+2\frac{5x-6}{(x-3)^2-14}$$
$$\text{II}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1-x}{1+x}=\frac{2}{1+x}-1$$
Look at II: the first fraction on the RHS must be an integer, so... The values you found with this substitute in I and do then some mathematics...
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