Combine the fractions on the left side:
9/(x + 2) + 2/(x - 2) = 1
9 (x - 2) / ((x + 2) (x - 2)) + 2 (x + 2) / ((x + 2) (x - 2)) = 1
(9x - 18) / ((x + 2) (x - 2)) + (2x + 4) / ((x + 2) (x - 2)) = 1
(9x - 18 + 2x + 4) / ((x + 2) (x - 2)) = 1
(11x - 14) / ((x + 2) (x - 2)) = 1
Recall the difference of squares identity,
a² - b² = (a + b) (a - b)
which lets us simplify the denominator on the left side as
(11x - 14) / (x² - 4) = 1
For any real numbers a and b, if a/b = 1, then a = b. This means
11x - 14 = x² - 4
which we can rearrange as
x² - 11x + 10 = 0
Factorize the left side:
(x - 10) (x - 1) = 0
Then
x - 10 = 0 or x - 1 = 0
x = 10 or x = 1