1. **State the problem:** Express the rational function $$\frac{x^2 + 2x - 8}{x^3 (x^4 - 2)}$$ in partial fractions. 2. **Identify the denominator factors:** The denominator is $$x^3 (x^4 - 2)$$. Here, $$x^3$$ is a repeated linear factor and $$x^4 - 2$$ is an irreducible quartic polynomial. 3. **Set up the partial fraction decomposition:** For the repeated linear factor $$x^3$$, we write terms with powers 1 to 3: $$\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3}$$ For the quartic polynomial $$x^4 - 2$$, since it is irreducible, we use a general linear numerator: $$\frac{Dx^3 + Ex^2 + Fx + G}{x^4 - 2}$$ 4. **Write the full decomposition:** $$\frac{x^2 + 2x - 8}{x^3 (x^4 - 2)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{Dx^3 + Ex^2 + Fx + G}{x^4 - 2}$$ 5. **Multiply both sides by the denominator to clear fractions:** $$x^2 + 2x - 8 = A x^2 (x^4 - 2) + B x (x^4 - 2) + C (x^4 - 2) + (Dx^3 + Ex^2 + Fx + G) x^3$$ 6. **Expand and collect like terms:** Expand each term and group powers of $$x$$ to form a polynomial identity. 7. **Equate coefficients of corresponding powers of $$x$$:** This will give a system of equations to solve for $$A, B, C, D, E, F, G$$. 8. **Solve the system:** Use substitution or matrix methods to find the values of the constants. 9. **Write the final partial fraction decomposition:** Substitute the constants back into the decomposition formula. This process expresses the given rational function as a sum of simpler fractions, which is useful for integration or further algebraic manipulation.