First notice that
x ² + 5x + 6 = (x + 3) (x + 2)
so you can rewrite everything with a common denominator:
5 / (x ² + 5x + 6) + 7 / (x + 2) = 6 / (x + 3)
5 / (x ² + 5x + 6) + 7 (x + 3) / ((x + 2) (x + 3)) = 6 (x + 2) / ((x + 3) (x + 2))
5 / (x ² + 5x + 6) + 7 (x + 3) / (x ² + 5x + 6) = 6 (x + 2) / (x ² + 5x + 6)
(5 + 7 (x + 3)) / (x ² + 5x + 6) = 6 (x + 2) / (x ² + 5x + 6)
As long as x ≠ -3 or x ≠ -2, the denominators are never zero, so we can cancel them and be left with
5 + 7 (x + 3) = 6 (x + 2)
Solving for x is easy from here:
5 + 7x + 21 = 6x + 12
7x - 6x = 12 - 5 - 21
x = -14