https://file.aiquickdraw.com/mathbot/file/1726982368873d709v6bf.jpg

I can see that the image contains a solution to a quadratic inequality and a corresponding table to check the values. I'll go through the steps displayed in the image and then verify the solution.

### Steps provided in the image:

1. **Inequality**: 
   \[
   x^2 + 2x - 3 \leq 0
   \]

2. **Factoring**:
   The quadratic expression is factored as:
   \[
   (x - 1)(x + 3) \leq 0
   \]

   The solutions to the equation are \(x = 1\) and \(x = -3\), which divide the number line into three sections: \(x < -3\), \(-3 \leq x \leq 1\), and \(x > 1\).

3. **Test values for each section**:
   - For \(x < -3\), a test value of \(x = -4\) is chosen.
   - For \(-3 \leq x \leq 1\), a test value of \(x = 0\) is chosen.
   - For \(x > 1\), a test value of \(x = 2\) is chosen.

4. **Results for each test value**:
   - When \(x = -4\): \( (-4 - 1)(-4 + 3) = (-5)(-1) = 5 \), which is positive, so this section does not satisfy the inequality.
   - When \(x = 0\): \( (0 - 1)(0 + 3) = (-1)(3) = -3 \), which is negative, so this section satisfies the inequality.
   - When \(x = 2\): \( (2 - 1)(2 + 3) = (1)(5) = 5 \), which is positive, so this section does not satisfy the inequality.

5. **Solution**:
   The inequality \( (x - 1)(x + 3) \leq 0 \) holds for the interval:
   \[
   -3 \leq x \leq 1
   \]

Thus, the solution to the inequality \(x^2 + 2x - 3 \leq 0\) is \(x \in [-3, 1]\).

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Do you need further details or clarification on this? Here are some additional related questions:

1. How do we know when to factor a quadratic equation?
2. What is the significance of test points in solving inequalities?
3. How do we interpret the inequality signs (\(\leq\), \(\geq\)) in solutions?
4. Can a quadratic inequality have no real solution? Why?
5. What is the difference between solving \(x^2 + 2x - 3 \leq 0\) and \(x^2 + 2x - 3 = 0\)?

### Tip:
Always test values from each section of the number line when solving inequalities. This helps in determining where the inequality holds true.

A detailed step-by-step solution to solve the quadratic inequality x^2 + 2x - 3 ≤ 0 using factoring and the test point method. Learn how to break down quadratic expressions and test intervals for truth values. Suitable for high school students in Grades 9-10.

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