I subbed your solution values into the equation the way you typed it and neither solution works
they do work if you meant
8/x - 1 = 9/(x(x+2)) ,
so I will start from there
multiply each term by x(x+2)
8(x+2) - x(x+2) = 9
8x + 16 - x^2 - 2x = 9
x^2 -6x -7 = 0
(x-7)(x+1) = 0
x = 7 or x = -1
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