9-a
look the polynomial 2x²+2x-4 has as roots 1 and -2 (if we replace x by 1 or -2 in the polynomial we'll get 0)
For that, we can be factorized, in an other word, the polynomial can be divided by (x-1) or (x+2)
2x²+2x-4=(x-1)(2x+4)
2x²+2x-4=(x+2)(2x-2)
\(\frac{1}{2}\frac{1}{(x+2)} + \frac{1}{x-1} - \frac{x+5}{2x²+2x-4} = \frac{1}{2}\frac{2x-2}{(x+2)(2x-2)} + \frac{2x+4}{(x-1)(2x+4)} - \frac{x+5}{2x²+2x-4} \)
\( = \frac{1}{2}\frac{2(x-1)}{(x+2)(2x-2)} + \frac{2x+4}{(x-1)(2x+4)} - \frac{x+5}{2x²+2x-4} \)
\( = \frac{2}{2}\frac{x-1}{(x+2)(2x-2)} + \frac{2x+4}{(x-1)(2x+4)} - \frac{x+5}{2x²+2x-4} \)
\( = \frac{x-1}{2x²+2x-4} + \frac{2x+4}{2x²+2x-4} - \frac{x+5}{2x²+2x-4} \)
\( = \frac{2x-2}{2x²+2x-4} = \frac{2(x-1)}{2x²+2x-4} = \frac{2(x-1)}{(x-1)(2x+4) }= \frac{2}{2(x+2) }\)
\(=\frac{1}{(x+2) }\)
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