To solve the quadratic equation \(2x^2 + 4x = -12\), we first rearrange it into standard form:
\[ 2x^2 + 4x + 12 = 0 \]
Next, we can simplify this equation by dividing all terms by 2:
\[ x^2 + 2x + 6 = 0 \]
Now, we will use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = 2\), and \(c = 6\).
Calculating the discriminant:
\[ b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 6 = 4 - 24 = -20 \]
Since the discriminant is negative, the solutions will involve complex numbers.
Now applying the quadratic formula:
\[ x = \frac{-2 \pm \sqrt{-20}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{-2 \pm \sqrt{20}i}{2} \]
We can simplify \(\sqrt{20}\):
\[ \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} \]
So we substitute this back into our formula:
\[ x = \frac{-2 \pm 2\sqrt{5}i}{2} = -1 \pm \sqrt{5}i \]
Therefore, the solutions can be expressed as:
\[ -1 \pm i\sqrt{5} \]
The correct answer is:
\(-1 \pm i\sqrt{5}\) which corresponds to the second option in your responses.