1. **State the problem:** Simplify the expression $$\frac{4a^2-(x-3)^2}{(2a+x)^2-9}$$. 2. **Recall formulas:** This expression involves differences of squares. Recall that $$A^2 - B^2 = (A-B)(A+B)$$. 3. **Apply difference of squares to numerator:** $$4a^2-(x-3)^2 = (2a)^2 - (x-3)^2 = (2a - (x-3))(2a + (x-3))$$ 4. **Apply difference of squares to denominator:** $$(2a+x)^2 - 9 = (2a+x)^2 - 3^2 = ((2a+x) - 3)((2a+x) + 3)$$ 5. **Rewrite numerator and denominator:** $$\frac{(2a - (x-3))(2a + (x-3))}{((2a+x) - 3)((2a+x) + 3)}$$ 6. **Simplify terms inside parentheses:** - Numerator: - $$2a - (x-3) = 2a - x + 3 = (2a + 3 - x)$$ - $$2a + (x-3) = 2a + x - 3$$ - Denominator: - $$(2a + x) - 3 = 2a + x - 3$$ - $$(2a + x) + 3 = 2a + x + 3$$ 7. **Rewrite expression with simplified terms:** $$\frac{(2a + 3 - x)(2a + x - 3)}{(2a + x - 3)(2a + x + 3)}$$ 8. **Cancel common factors:** The term $$2a + x - 3$$ appears in numerator and denominator, so cancel it: $$\frac{2a + 3 - x}{2a + x + 3}$$ 9. **Final simplified expression:** $$\boxed{\frac{2a + 3 - x}{2a + x + 3}}$$ This is the simplified form of the original expression.