To solve the equation \(-4x(-x + 6) = -8\), let's follow these steps:

1. **Distribute** \(-4x\) through the parentheses:
   \[
   -4x \cdot (-x) + (-4x) \cdot 6 = -8
   \]

2. **Simplify** the terms:
   \[
   4x^2 - 24x = -8
   \]

3. **Move all terms** to one side to set the equation to 0:
   \[
   4x^2 - 24x + 8 = 0
   \]

4. **Divide the entire equation** by 4 to simplify:
   \[
   x^2 - 6x + 2 = 0
   \]

5. **Solve the quadratic equation** using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -6\), and \(c = 2\):
   \[
   x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}
   \]
   \[
   x = \frac{6 \pm \sqrt{36 - 8}}{2}
   \]
   \[
   x = \frac{6 \pm \sqrt{28}}{2}
   \]
   \[
   x = \frac{6 \pm 2\sqrt{7}}{2}
   \]
   \[
   x = 3 \pm \sqrt{7}
   \]

So, the solutions to the equation \(-4x(-x + 6) = -8\) are:
\[
x = 3 + \sqrt{7} \quad \text{and} \quad x = 3 - \sqrt{7}
\]

Would you like more details or have any questions?

Here are 8 related questions:
1. How do you apply the distributive property in algebraic expressions?
2. What are the steps to solve a quadratic equation?
3. How do you simplify the expression \(\sqrt{28}\)?
4. What is the quadratic formula and how is it derived?
5. Can you solve the quadratic equation \(x^2 - 4x + 4 = 0\)?
6. How do you complete the square to solve a quadratic equation?
7. What are the properties of the solutions to a quadratic equation?
8. How do you factorize the quadratic expression \(x^2 - 6x + 2\)?

**Tip:** When solving quadratic equations, always double-check your solutions by plugging them back into the original equation to ensure they satisfy it.

Learn how to solve the equation -4x(-x + 6) = -8 with detailed steps and explanations. This problem involves distributing, simplifying, and solving a quadratic equation using the quadratic formula. Suitable for high school students studying algebra and quadratic equations.

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