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).

Learn how to solve the quadratic equation -2x^2 + 2x + 8 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the quadratic formula and solving quadratic equations. Suitable for students in Grades 9-12.

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