Solve the Quadratic Equation: (x+2)^2 = (2x+3)^2

We will solve the equation $(x+2)^2 = (2x+3)^2$ by expanding and simplifying both sides:

Step 1: Expand both sides of the equation:
Left side: $(x+2)^2 = x^2 + 4x + 4$
Right side: $(2x+3)^2 = 4x^2 + 12x + 9$

Step 2: Set the expanded forms equal to each other:
$x^2 + 4x + 4 = 4x^2 + 12x + 9$

Step 3: Rearrange to form a standard quadratic equation:
Subtract $x^2 + 4x + 4$ from both sides:
$0 = 3x^2 + 8x + 5$

Step 4: Rearrange to get:
$3x^2 + 8x + 5 = 0$

Step 5: Solve using the quadratic formula:
Using $a = 3$, $b = 8$, $c = 5$:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
$x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 3 \cdot 5}}{2 \cdot 3}$
$x = \frac{-8 \pm \sqrt{64 - 60}}{6}$
$x = \frac{-8 \pm \sqrt{4}}{6}$
$x = \frac{-8 \pm 2}{6}$

Step 6: Calculate the solutions:
$x_1 = \frac{-8 + 2}{6} = \frac{-6}{6} = -1$
$x_2 = \frac{-8 - 2}{6} = \frac{-10}{6} = -\frac{5}{3}$

Verify in the original equation to assure correctness. Hence, both solutions are valid.

Therefore, the solutions are $x_1 = -1$ and $x_2 = -\frac{5}{3}$, which matches choice 3.