To solve the quadratic equation \( 2x^2 + 4x = -12 \), we first rewrite it in standard form:
\[ 2x^2 + 4x + 12 = 0 \]
Next, we can divide the entire equation by 2 to simplify:
\[ x^2 + 2x + 6 = 0 \]
Now, we'll 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, we will have complex solutions. Now we substitute into the quadratic formula:
\[ x = \frac{-2 \pm \sqrt{-20}}{2 \cdot 1} = \frac{-2 \pm \sqrt{20}i}{2} \]
We can simplify \(\sqrt{20}\):
\[ \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} \]
So we substitute this back into our expression:
\[ x = \frac{-2 \pm 2\sqrt{5} i}{2} = -1 \pm \sqrt{5} i \]
Thus, the solutions in the form \( a \pm bi \) are:
\[ -1 \pm i\sqrt{5} \]
From the options given, the correct response from your choices is:
\(-1 \pm i\sqrt{5}\).