1. **State the problem:** Simplify the expression $$\frac{3}{x^2 - 4} + \frac{1}{(x-2)^2}$$. 2. **Recall the formula and rules:** Recognize that $$x^2 - 4$$ is a difference of squares and can be factored as $$x^2 - 4 = (x-2)(x+2)$$. 3. **Rewrite the expression using the factorization:** $$\frac{3}{(x-2)(x+2)} + \frac{1}{(x-2)^2}$$ 4. **Find the common denominator:** The least common denominator (LCD) is $$ (x-2)^2 (x+2) $$. 5. **Rewrite each fraction with the LCD:** - First fraction: multiply numerator and denominator by $$x-2$$ to get $$\frac{3(x-2)}{(x-2)^2 (x+2)}$$. - Second fraction: multiply numerator and denominator by $$x+2$$ to get $$\frac{1(x+2)}{(x-2)^2 (x+2)}$$. 6. **Add the numerators over the common denominator:** $$\frac{3(x-2) + (x+2)}{(x-2)^2 (x+2)}$$ 7. **Simplify the numerator:** $$3(x-2) + (x+2) = 3x - 6 + x + 2 = 4x - 4$$ 8. **Factor the numerator:** $$4x - 4 = 4(x - 1)$$ 9. **Write the final simplified expression:** $$\frac{4(x-1)}{(x-2)^2 (x+2)}$$ **Answer:** $$\frac{4(x-1)}{(x-2)^2 (x+2)}$$