Find the real solutions, if any, of the following equation. Use the quadratic formula. $$ 2 x(x+2)=4 $$ Select the correct choice below and, if necessary, fill in the answer box to complete your answer. A. The solution set is $$ \} $$ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) B. The solution is not a real number.

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To solve the equation $$ 2x(x + 2) = 4 $$ using the quadratic formula, follow these steps:

1. **Expand and Rearrange the Equation:**
   $$
   2x(x + 2) = 4 \
   2x^2 + 4x - 4 = 0 \
   \text{Divide by 2:} \quad x^2 + 2x - 2 = 0
   $$

2. **Identify the Coefficients:**
   $$
   a = 1, \quad b = 2, \quad c = -2
   $$

3. **Apply the Quadratic Formula:**
   $$
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \
   x = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-2)}}{2(1)} \
   x = \frac{-2 \pm \sqrt{4 + 8}}{2} \
   x = \frac{-2 \pm \sqrt{12}}{2} \
   x = \frac{-2 \pm 2\sqrt{3}}{2} \
   x = -1 \pm \sqrt{3}
   $$

4. **Determine the Real Solutions:**
   The discriminant ($$ b^2 - 4ac = 12 $$) is positive, indicating two real solutions:
   $$
   x = -1 + \sqrt{3} \quad \text{and} \quad x = -1 - \sqrt{3}
   $$

**Final Answer:**

**A. The solution set is $$\{-1 - \sqrt{3}, -1 + \sqrt{3}\}$$**
The solutions are $$ x = -1 - \sqrt{3} $$ and $$ x = -1 + \sqrt{3} $$.

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