[tex](\frac{x}{x^{2} -8x+16} - \frac{x+6}{x^{2} -16}) : \frac{x+12}{x^{2} -16} = (\frac{x}{(x-4)^{2} } - \frac{x+6}{(x-4)(x+4)}) : \frac{x+12}{x^{2} -16} = \frac{x(x+4)-(x-4)(x+6)}{(x-4)^2(x+4)} *\frac{x^{2}-16 }{x+12} = \frac{x^2+4x-(x^2+6x-4x-24)}{(x-4)^2(x+4)} * \frac{x^2-16}{x+12} = \frac{x(x+4)-(x-4)(x+6)}{(x-4)^2(x+4)} *\frac{x^{2}-16 }{x+12} = \frac{x^2+4x-(x^2+2x-24)}{(x-4)^2(x+4)} * \frac{x^2-16}{x+12} = \frac{x^2+4x-x^2-2x+24}{(x-4)^2(x+4)} * \frac{x^2-16}{x+12} =[/tex][tex]\frac{x^2+4x-x^2-2x+24}{(x-4)^2(x+4)} * \frac{x^2-16}{x+12} = \frac{2x+24}{(x-4)^2(x+4)} * \frac{x^2-16}{x+12} = \frac{2(x+12)}{(x-4)^2(x+4)} * \frac{(x-4)(x+4)}{x+12} = \frac{2}{x-4}[/tex]
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[tex](\frac{x}{x^{2} -8x+16} - \frac{x+6}{x^{2} -16}) : \frac{x+12}{x^{2} -16} = (\frac{x}{(x-4)^{2} } - \frac{x+6}{(x-4)(x+4)}) : \frac{x+12}{x^{2} -16} = \frac{x(x+4)-(x-4)(x+6)}{(x-4)^2(x+4)} *\frac{x^{2}-16 }{x+12} = \frac{x^2+4x-(x^2+6x-4x-24)}{(x-4)^2(x+4)} * \frac{x^2-16}{x+12} = \frac{x(x+4)-(x-4)(x+6)}{(x-4)^2(x+4)} *\frac{x^{2}-16 }{x+12} = \frac{x^2+4x-(x^2+2x-24)}{(x-4)^2(x+4)} * \frac{x^2-16}{x+12} = \frac{x^2+4x-x^2-2x+24}{(x-4)^2(x+4)} * \frac{x^2-16}{x+12} =[/tex][tex]\frac{x^2+4x-x^2-2x+24}{(x-4)^2(x+4)} * \frac{x^2-16}{x+12} = \frac{2x+24}{(x-4)^2(x+4)} * \frac{x^2-16}{x+12} = \frac{2(x+12)}{(x-4)^2(x+4)} * \frac{(x-4)(x+4)}{x+12} = \frac{2}{x-4}[/tex]