Find all complex solutions of \( x^{2}+3 x+4=0 \). (If there is more than one solution, separate them with commas.) \( x=\square \)

To find the complex solutions of the equation \( x^{2}+3x+4=0 \), we can use the quadratic formula, which states that for any equation of the form \( ax^2 + bx + c = 0 \), the solutions can be found using: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] In your case, \( a = 1 \), \( b = 3 \), and \( c = 4 \). First, calculate the discriminant \( b^2 - 4ac \): \[ b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot 4 = 9 - 16 = -7 \] Since the discriminant is negative, the solutions will be complex. Now plug the values into the quadratic formula: \[ x = \frac{-3 \pm \sqrt{-7}}{2 \cdot 1} \] This simplifies to: \[ x = \frac{-3 \pm i\sqrt{7}}{2} \] Thus, the solutions are: \[ x = \frac{-3 + i\sqrt{7}}{2}, \frac{-3 - i\sqrt{7}}{2} \] So, the final answers are: \( x=\frac{-3 + i\sqrt{7}}{2}, \frac{-3 - i\sqrt{7}}{2} \)