https://file.aiquickdraw.com/mathbot/file/1729101739421k020jb4t.jpg

The equation provided in the image is:

\[
-4(x - 5)(2x + 5) = 0
\]

To solve this using the Zero-Factor Principle, we can break it down as follows:

### Step 1: Zero-Factor Principle
The equation has the form:
\[
A \cdot B = 0
\]
which implies that either \( A = 0 \) or \( B = 0 \).

Here, \( A = -4 \) and \( B = (x - 5)(2x + 5) \). Since \( -4 \) is a constant and cannot be zero, we focus on the part involving the variable \( x \):
\[
(x - 5)(2x + 5) = 0
\]

### Step 2: Solve each factor
Set each factor equal to zero and solve for \( x \).

1. \( x - 5 = 0 \)
   \[
   x = 5
   \]

2. \( 2x + 5 = 0 \)
   \[
   2x = -5
   \]
   \[
   x = -\frac{5}{2}
   \]

### Step 3: Final answer
The solutions are:
\[
x = 5 \quad \text{or} \quad x = -\frac{5}{2}
\]

These are the two values of \( x \) that satisfy the equation.

---

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

Here are 5 questions related to the problem:
1. How would the solution change if the constant were something other than -4?
2. Can you explain why we ignore the constant -4 in the Zero-Factor Principle?
3. What happens if one of the factors involves a square, like \( (x - 5)^2 \)?
4. What does it mean when an equation has no solution?
5. Can you use factoring when the equation has more complex terms like \( (x - 5)(3x^2 + 2x - 4) \)?

**Tip:** Always check if the equation can be simplified before solving, as it may make the factoring easier.

Solve the given equation using the Zero-Factor Principle. Present your answer as an integer or reduced fraction. If there is more than one solution, separate them with a comma. If there is no solution, enter DNE. -4(x - 5)(2x + 5) = 0

Learn how to solve the equation -4(x - 5)(2x + 5) = 0 using the Zero-Factor Principle. This step-by-step guide covers factoring and solving for variable values, making it ideal for students in Grades 8-10.

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