1. **State the problem:** We want to simplify the expression $$\frac{5x^2}{x^4 - 1}$$ and express it in partial fractions if possible. 2. **Factor the denominator:** Notice that $$x^4 - 1$$ is a difference of squares: $$x^4 - 1 = (x^2)^2 - 1^2 = (x^2 - 1)(x^2 + 1)$$ 3. Further factor the quadratic difference of squares: $$x^2 - 1 = (x - 1)(x + 1)$$ 4. So the denominator factors as: $$x^4 - 1 = (x - 1)(x + 1)(x^2 + 1)$$ 5. **Set up partial fractions:** $$\frac{5x^2}{(x - 1)(x + 1)(x^2 + 1)} = \frac{A}{x - 1} + \frac{B}{x + 1} + \frac{Cx + D}{x^2 + 1}$$ 6. Multiply both sides by the denominator to clear fractions: $$5x^2 = A(x + 1)(x^2 + 1) + B(x - 1)(x^2 + 1) + (Cx + D)(x - 1)(x + 1)$$ 7. Expand each term: - $$A(x + 1)(x^2 + 1) = A(x^3 + x^2 + x + 1)$$ - $$B(x - 1)(x^2 + 1) = B(x^3 - x^2 + x - 1)$$ - $$ (Cx + D)(x^2 - 1) = Cx(x^2 - 1) + D(x^2 - 1) = C(x^3 - x) + D(x^2 - 1)$$ 8. Combine all terms: $$5x^2 = A(x^3 + x^2 + x + 1) + B(x^3 - x^2 + x - 1) + C(x^3 - x) + D(x^2 - 1)$$ 9. Group like terms: $$5x^2 = (A + B + C)x^3 + (A - B + D)x^2 + (A + B - C)x + (A - B - D)$$ 10. Equate coefficients of powers of $$x$$ on both sides: - Coefficient of $$x^3$$: $$0 = A + B + C$$ - Coefficient of $$x^2$$: $$5 = A - B + D$$ - Coefficient of $$x$$: $$0 = A + B - C$$ - Constant term: $$0 = A - B - D$$ 11. Solve the system: From the first and third equations: $$0 = A + B + C$$ $$0 = A + B - C$$ Add both: $$0 = 2(A + B) \Rightarrow A + B = 0$$ Then from first: $$C = - (A + B) = 0$$ From last two equations: $$5 = A - B + D$$ $$0 = A - B - D$$ Add both: $$5 = 2(A - B) \Rightarrow A - B = \frac{5}{2}$$ From $$A + B = 0$$, we get $$B = -A$$. Substitute into $$A - B = \frac{5}{2}$$: $$A - (-A) = 2A = \frac{5}{2} \Rightarrow A = \frac{5}{4}$$ Then $$B = -\frac{5}{4}$$. From $$0 = A - B - D$$: $$D = A - B = \frac{5}{4} - (-\frac{5}{4}) = \frac{5}{2}$$ 12. **Final partial fraction decomposition:** $$\frac{5x^2}{x^4 - 1} = \frac{5/4}{x - 1} - \frac{5/4}{x + 1} + \frac{0 \cdot x + 5/2}{x^2 + 1} = \frac{5}{4(x - 1)} - \frac{5}{4(x + 1)} + \frac{5/2}{x^2 + 1}$$ **Answer:** $$\boxed{\frac{5}{4(x - 1)} - \frac{5}{4(x + 1)} + \frac{5/2}{x^2 + 1}}$$