1. **State the problem:** Calculate the value of the complex number expression $ (2 + 3i)^2 $. 2. **Formula and rules:** To square a complex number $ (a + bi) $, use the formula: $$ (a + bi)^2 = a^2 + 2abi + (bi)^2 = a^2 - b^2 + 2abi $$ Note that $ i^2 = -1 $. 3. **Apply the formula:** Here, $ a = 2 $ and $ b = 3 $. $$ (2 + 3i)^2 = 2^2 + 2 \times 2 \times 3i + (3i)^2 $$ 4. **Calculate each term:** $$ 2^2 = 4 $$ $$ 2 \times 2 \times 3i = 12i $$ $$ (3i)^2 = 3^2 \times i^2 = 9 \times (-1) = -9 $$ 5. **Combine terms:** $$ 4 + 12i - 9 = (4 - 9) + 12i = -5 + 12i $$ **Final answer:** $$ (2 + 3i)^2 = -5 + 12i $$