1. The problem is to perform partial fraction decomposition on a given rational expression. 2. Partial fraction decomposition is used to express a rational function as a sum of simpler fractions, which is useful for integration and other operations. 3. Suppose we have a rational function $$\frac{P(x)}{Q(x)}$$ where the degree of $$P(x)$$ is less than the degree of $$Q(x)$$ and $$Q(x)$$ can be factored into linear or irreducible quadratic factors. 4. The general form of partial fractions depends on the factorization of $$Q(x)$$. For example, if $$Q(x) = (x-a)(x-b)$$, then: $$\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$$ 5. To find constants $$A$$ and $$B$$, multiply both sides by $$Q(x)$$ to clear denominators: $$P(x) = A(x-b) + B(x-a)$$ 6. Expand and collect like terms, then equate coefficients of powers of $$x$$ on both sides to form a system of equations. 7. Solve the system for $$A$$ and $$B$$. 8. Substitute back to write the original expression as the sum of partial fractions. 9. Example: Decompose $$\frac{3x+5}{(x-1)(x+2)}$$. 10. Set $$\frac{3x+5}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$$. 11. Multiply both sides by $$(x-1)(x+2)$$: $$3x+5 = A(x+2) + B(x-1)$$ 12. Expand: $$3x+5 = Ax + 2A + Bx - B = (A+B)x + (2A - B)$$ 13. Equate coefficients: For $$x$$: $$3 = A + B$$ For constant: $$5 = 2A - B$$ 14. Solve the system: From $$3 = A + B$$, $$B = 3 - A$$ Substitute into $$5 = 2A - B$$: $$5 = 2A - (3 - A) = 2A - 3 + A = 3A - 3$$ $$3A = 8$$ $$A = \frac{8}{3}$$ 15. Then $$B = 3 - \frac{8}{3} = \frac{9}{3} - \frac{8}{3} = \frac{1}{3}$$ 16. Final decomposition: $$\frac{3x+5}{(x-1)(x+2)} = \frac{8/3}{x-1} + \frac{1/3}{x+2}$$ 17. This completes the partial fraction decomposition.