To solve the equation:

$x^2 + x = -2$

we start by rearranging it to set it equal to zero:

$x^2 + x + 2 = 0$

Now, we need to determine if this quadratic equation has real solutions by using the discriminant in the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case:

• $a = 1$
• $b = 1$
• $c = 2$

1. Calculate the discriminant:

   $b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot 2 = 1 - 8 = -7$

Since the discriminant is negative ($-7$), there are no real solutions for this equation. The solutions are complex.

2. Finding the complex solutions:

   $x = \frac{-1 \pm \sqrt{-7}}{2 \cdot 1} = \frac{-1 \pm \sqrt{7}i}{2}$

So, the solutions are:

$x = \frac{-1 + \sqrt{7}i}{2} \quad \text{and} \quad x = \frac{-1 - \sqrt{7}i}{2}$

These are the complex solutions to the equation.

Would you like more details on any step or have further questions?

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Here are some related questions you might find interesting:

1. How do we determine if a quadratic equation has real or complex solutions?
2. What are imaginary numbers, and why do we use $i$ to represent $\sqrt{-1}$?
3. How can we graph complex solutions on the complex plane?
4. What does the discriminant tell us about the nature of solutions?
5. How can we factor quadratic equations with complex solutions?

Tip: When working with quadratic equations, the discriminant $b^2 - 4ac$ quickly reveals the type of solutions you'll encounter: positive for real and distinct, zero for a double root, and negative for complex roots.