https://file.aiquickdraw.com/mathbot/file/1726410460460g6e2o1w6.jpg

The problem shown in the image is a quadratic inequality, and the equation given is:

\[
4x^2 + 6x = 2x(x + 3)
\]

### Step 1: Expand and simplify the inequality

First, let's simplify the equation:

\[
4x^2 + 6x = 2x(x + 3)
\]

Expand the right-hand side:

\[
4x^2 + 6x = 2x^2 + 6x
\]

Now subtract \(2x^2 + 6x\) from both sides to simplify:

\[
(4x^2 + 6x) - (2x^2 + 6x) = 0
\]

Simplifying further:

\[
2x^2 = 0
\]

Thus, \(x = 0\).

### Step 2: Analyze the solution in terms of intervals

We need to analyze the behavior of the quadratic equation around the critical point \(x = 0\). 

Since \(x = 0\) is a solution, we will test the intervals on either side of \(x = 0\).

The intervals are:

- \( (-\infty, 0) \)
- \( (0, \infty) \)

### Step 3: Create the table

| Testing Interval | Interval Notation | Testing Number | Conclusion |
|------------------|-------------------|----------------|------------|
| \( (-\infty, 0) \) | \( (-\infty, 0) \) | \( x = -1 \) | |
| \( (0, \infty) \) | \( (0, \infty) \) | \( x = 1 \) | |

#### Testing the intervals:
- For \(x = -1\) in \( (-\infty, 0) \):
  \[
  4(-1)^2 + 6(-1) = 4(1) - 6 = -2
  \]
  and 
  \[
  2(-1)(-1 + 3) = 2(-1)(2) = -4
  \]

- For \(x = 1\) in \( (0, \infty) \):
  \[
  4(1)^2 + 6(1) = 4 + 6 = 10
  \]
  and 
  \[
  2(1)(1 + 3) = 2(1)(4) = 8
  \]

### Step 4: Fill in the Conclusion

Based on these calculations, the table will be updated with conclusions based on the values for the respective intervals.

Do you want further details on any specific step or a different approach? Let me know if you'd like to refine this solution further!

Here are 5 related questions you could explore:
1. How do you solve a quadratic inequality algebraically?
2. What is the significance of finding critical points in solving inequalities?
3. Can a quadratic inequality have multiple solutions?
4. How do you determine the intervals to test when solving inequalities?
5. What is the relationship between the discriminant and the number of solutions for a quadratic equation?

**Tip**: When solving quadratic inequalities, always analyze the sign of the expression in each interval between critical points to determine the solution set.

Solve the quadratic inequality 4x^2 + 6x = 2x(x + 3) using interval testing and complete the table.

This page provides a step-by-step solution to the quadratic inequality 4x^2 + 6x = 2x(x + 3) using the interval testing method. The solution involves simplifying the equation, identifying critical points, and testing values in different intervals.

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