Question: Solve the following equations using any method you know. Show all work. Write your final answer(s) as a solution set.
$(2x+3)^2 -9+0$
1. **State the problem:** Solve the equation $$ (2x+3)^2 - 9 = 0 $$.
2. **Rewrite the equation:**
$$ (2x+3)^2 - 9 = 0 $$
3. **Recognize the difference of squares:**
$$ a^2 - b^2 = (a-b)(a+b) $$ where $$ a = 2x+3 $$ and $$ b = 3 $$.
4. **Factor the equation:**
$$ ((2x+3) - 3)((2x+3) + 3) = 0 $$
5. **Simplify each factor:**
$$ (2x + 3 - 3)(2x + 3 + 3) = (2x)(2x + 6) = 0 $$
6. **Set each factor equal to zero:**
- $$ 2x = 0 $$
- $$ 2x + 6 = 0 $$
7. **Solve each equation:**
- For $$ 2x = 0 $$:
$$ x = \frac{\cancel{2}x}{\cancel{2}} = 0 $$
- For $$ 2x + 6 = 0 $$:
$$ 2x = -6 $$
$$ x = \frac{-6}{2} = -3 $$
8. **Write the solution set:**
$$ \boxed{\{0, -3\}} $$