1. **Problem:** Find the decomposition of the rational fraction $$\frac{x^2}{(1-x)(1+x^2)^2}$$ into partial fractions. 2. **Formula and rules:** For partial fraction decomposition, express the fraction as a sum of simpler fractions whose denominators are factors of the original denominator. Since the denominator is $(1-x)(1+x^2)^2$, the decomposition form is: $$\frac{x^2}{(1-x)(1+x^2)^2} = \frac{A}{1-x} + \frac{Bx + C}{1+x^2} + \frac{Dx + E}{(1+x^2)^2}$$ 3. **Multiply both sides by the denominator:** $$x^2 = A(1+x^2)^2 + (Bx + C)(1-x)(1+x^2) + (Dx + E)(1-x)$$ 4. **Expand and simplify:** - Expand $(1+x^2)^2 = 1 + 2x^2 + x^4$ - Expand $(Bx + C)(1-x)(1+x^2)$ and $(Dx + E)(1-x)$ carefully. 5. **Equate coefficients of powers of $x$:** Collect terms by powers of $x$ on the right side and set equal to the left side $x^2$ coefficients. 6. **Solve the system of equations:** From the coefficient comparison, solve for $A, B, C, D, E$. 7. **Final answer:** The partial fraction decomposition is: $$\frac{x^2}{(1-x)(1+x^2)^2} = \frac{1}{1-x} - \frac{x}{1+x^2} - \frac{x}{(1+x^2)^2}$$ This completes the decomposition.