To solve the equation \(-6x^2 - 3x - 2 = -8\), let's follow these steps:

### Step 1: Move all terms to one side of the equation.
Add \(8\) to both sides to get:

\[
-6x^2 - 3x - 2 + 8 = 0
\]

Simplify the equation:

\[
-6x^2 - 3x + 6 = 0
\]

### Step 2: Simplify the equation (if possible).
Divide the entire equation by \(-3\) to make the coefficients smaller:

\[
2x^2 + x - 2 = 0
\]

### Step 3: Solve the quadratic equation.
We can solve this quadratic equation using the quadratic formula:

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

Here, \(a = 2\), \(b = 1\), and \(c = -2\). Plug these values into the formula:

\[
x = \frac{-1 \pm \sqrt{1^2 - 4(2)(-2)}}{2(2)}
\]
\[
x = \frac{-1 \pm \sqrt{1 + 16}}{4}
\]
\[
x = \frac{-1 \pm \sqrt{17}}{4}
\]

### Step 4: Simplify the solutions.
The solutions are:

\[
x = \frac{-1 + \sqrt{17}}{4} \quad \text{or} \quad x = \frac{-1 - \sqrt{17}}{4}
\]

So, the solutions to the equation are:

\[
x = \frac{-1 + \sqrt{17}}{4} \quad \text{and} \quad x = \frac{-1 - \sqrt{17}}{4}
\]

### Final Answer:
\[
x = \frac{-1 + \sqrt{17}}{4} \quad \text{or} \quad x = \frac{-1 - \sqrt{17}}{4}
\]

Would you like further details or explanations?

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Here are some related questions you might find interesting:
1. How do you determine the nature of the roots of a quadratic equation?
2. What are some alternative methods to solve quadratic equations?
3. How does the discriminant of a quadratic equation affect the solutions?
4. Can you factor the quadratic equation \(2x^2 + x - 2 = 0\) instead of using the quadratic formula?
5. How do you solve a quadratic equation by completing the square?

**Tip:** When solving quadratic equations, always check if the equation can be simplified or factored before using the quadratic formula—it can save time and effort!

Learn how to solve the quadratic equation -6x^2 - 3x - 2 = -8 step-by-step with detailed solutions and explanations. This problem involves algebraic manipulation and the quadratic formula. Suitable for high school students in Grades 10-12.

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