#### Answer

$x = -2 \frac{+}{} \sqrt{2}i$

#### Work Step by Step

$$x^2 + 4x + 6 = 0$$
Since this is a quadratic equation, and it cannot be factorized in a simple manner, we can use the Quadratic Formula to solve for $x$: $$x = \frac{-b \frac{+}{} \sqrt{b^2 - 4ac}}{2a}$$ $$x = \frac{-(4) \frac{+}{} \sqrt{(4)^2 - 4(1)(6)}}{2(1)}$$ $$\frac{-4 \frac{+}{} \sqrt{-8}}{2}$$ Since we've arrived at a negative value inside a square root, we can say that $x$ has no solution within the realm of Real numbers. We CAN, however, solve this within the realm of the Universe of numbers by resorting to imaginary numbers: $$\frac{-4\frac{+}{} \sqrt{8}i}{2}$$ $$\frac{-4\frac{+}{}2\sqrt{2}i}{2}$$ $$-2 \frac{+}{} 2\sqrt{2}i$$