Let's solve the equation $8 + 2x = -2x^2$.

Step 1: Move all terms to one side of the equation

To set the equation to zero, we move all the terms to one side: $-2x^2 - 2x + 8 = 0$ We can rearrange it as: $-2x^2 - 2x + 8 = 0$

Step 2: Simplify the equation

We can divide the entire equation by $-2$ to simplify it: $x^2 + x - 4 = 0$

Step 3: Solve the quadratic equation

Now, we'll solve the quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ For our equation $x^2 + x - 4 = 0$, $a = 1$, $b = 1$, and $c = -4$.

Substitute these values into the quadratic formula: $x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-4)}}{2(1)}$ $x = \frac{-1 \pm \sqrt{1 + 16}}{2}$ $x = \frac{-1 \pm \sqrt{17}}{2}$

Step 4: Write the final solutions

The solutions to the equation are: $x = \frac{-1 + \sqrt{17}}{2} \quad \text{or} \quad x = \frac{-1 - \sqrt{17}}{2}$

These are the exact solutions. If you need approximate decimal values: $x \approx 1.56 \quad \text{or} \quad x \approx -2.56$

Would you like more details or have any questions about this solution?

Here are some related questions you might find interesting:

1. How can you verify the solutions of a quadratic equation?
2. What is the significance of the discriminant in the quadratic formula?
3. How do you factorize a quadratic equation?
4. What are the steps for completing the square for a quadratic equation?
5. Can every quadratic equation be solved using the quadratic formula?
6. What are the possible types of solutions for a quadratic equation?
7. How does the graph of a quadratic function relate to its solutions?
8. How can you determine the vertex of a quadratic equation?

Tip: Always check the discriminant $\Delta = b^2 - 4ac$ before solving a quadratic equation, as it tells you the nature of the roots (real and distinct, real and equal, or complex).