Question \# 24 / Page 668:
Write the partial fraction decomposition of the rational expression $\frac{x+2}{x\left(x^{2}-9\right)}$.
Solution:

Step 1 ：The given rational expression is a proper fraction, so we can directly proceed to write the partial fraction decomposition.

Step 2 ：The denominator factors as \(x(x-3)(x+3)\), so the partial fraction decomposition will have the form \(\frac{A}{x} + \frac{B}{x-3} + \frac{C}{x+3}\).

Step 3 ：We can find the values of A, B, and C by equating the original rational expression with the partial fraction decomposition and solving the resulting system of equations.

Step 4 ：The solution to the system of equations gives us \(A = -\frac{2}{9}\), \(B = \frac{5}{18}\), and \(C = -\frac{1}{18}\).

Step 5 ：Substituting these values back into the partial fraction decomposition gives us the final answer.

Step 6 ：\(\boxed{\frac{-2}{9x} + \frac{5}{18(x-3)} - \frac{1}{18(x+3)}}\)