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 the equation by dividing all terms by 2:
\[ x^2 + 2x + 6 = 0 \]
Now we can use the quadratic formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\), where \(a = 1\), \(b = 2\), and \(c = 6\).
Calculate the discriminant (\(b^2 - 4ac\)):
\[ b^2 - 4ac = 2^2 - 4(1)(6) = 4 - 24 = -20 \]
Since the discriminant is negative, we will have complex solutions. Now, we compute the solutions:
\[ x = \frac{{-2 \pm \sqrt{{-20}}}}{{2(1)}} \] \[ x = \frac{{-2 \pm \sqrt{{20}}i}}{{2}} \] \[ x = \frac{{-2 \pm 2\sqrt{5}i}}{{2}} \] \[ x = -1 \pm \sqrt{5}i \]
Thus, the solutions in the form \(a \pm bi\) are:
\[ -1 \pm i\sqrt{5} \]
From the options provided, the solution that matches is:
\[ \boxed{-1 \pm i\sqrt{5}} \]