1. Given the expression to decompose: $$x + \frac{3}{(x - 2)^2}$$ 2. Since the denominator is $(x - 2)^2$, consider partial fractions of the form: $$\frac{A}{x - 2} + \frac{B}{(x - 2)^2}$$ 3. Write the original fraction as: $$x + \frac{3}{(x - 2)^2} = \frac{A}{x - 2} + \frac{B}{(x - 2)^2}$$ 4. Multiply both sides by $(x - 2)^2$ to clear the denominator: $$ (x + \frac{3}{(x - 2)^2}) (x - 2)^2 = A(x - 2) + B $$ 5. Simplify left side: $$ x(x - 2)^2 + 3 = A(x - 2) + B $$ 6. Expand $(x - 2)^2$: $$ (x - 2)^2 = x^2 - 4x + 4 $$ 7. So left side becomes: $$ x(x^2 - 4x + 4) + 3 = x^3 - 4x^2 + 4x + 3 $$ 8. Set the equality: $$ x^3 - 4x^2 + 4x + 3 = A(x - 2) + B $$ This is not possible because left side is cubic and right side is linear in x, so the original expression is a polynomial plus a fraction, not a pure rational function suitable for partial fraction decomposition. 9. The partial fraction decomposition actually applies to proper rational functions. The expression $x + \frac{3}{(x - 2)^2}$ is already a sum of a polynomial $x$ and a proper rational fraction $\frac{3}{(x - 2)^2}$. 10. Therefore, the given expression cannot be decomposed further into simpler partial fractions because $x$ is not part of the fraction decomposition. **Final answer:** The expression is already decomposed: $$x + \frac{3}{(x - 2)^2}$$