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 \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = 2 \), and \( c = 6 \).
Calculating the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 6 = 4 - 24 = -20 \]
Since the discriminant is negative, we will have complex solutions.
Now we can plug in the values into the quadratic formula:
\[ x = \frac{-2 \pm \sqrt{-20}}{2 \cdot 1} \] \[ x = \frac{-2 \pm \sqrt{20}i}{2} \]
We can simplify \( \sqrt{20} \):
\[ \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} \]
Substituting back into the equation gives:
\[ x = \frac{-2 \pm 2\sqrt{5}i}{2} \] \[ x = -1 \pm \sqrt{5}i \]
Thus, the solutions can be expressed in the form \( -1 \pm i\sqrt{5} \).
The correct response is:
\(-1 \pm i\sqrt{5}\) (which can be written as \( -1 \pm i\sqrt{5} \)).