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The image you uploaded shows mathematical work related to solving quadratic equations and testing inequalities within specified intervals. The work is broken down into the following key sections:

1. **Quadratic Equations**:
   - \( x^2 + 3x - 40 = 0 \)
   - \( x^2 - 3x - 40 = 0 \)
   - \( x^2 - 8x + 112 = 0 \)

2. **Solution Sections**:
   - The intervals given are \( x < -1 \), \( -1 < x < 3 \), and \( x > 3 \).

3. **Testing the Inequality**:
   - A test is performed for the inequality \( x^2 + 3x - 40 \leq 0 \) using values from the intervals: \( x = -2 \), \( x = 0 \), and \( x = 4 \).

### Analysis:
- The quadratic equations need solving to find roots that help establish intervals where the inequality holds true.
- It appears tests have been conducted to verify the sign of the inequality within these intervals.

Let's solve these quadratic equations first and then determine the intervals where the inequality holds true.

#### Quadratic Equations:

1. \( x^2 + 3x - 40 = 0 \)
2. \( x^2 - 3x - 40 = 0 \)
3. \( x^2 - 8x + 112 = 0 \)

Let's solve each one to find their roots.The solutions to the quadratic equations are as follows:

1. \( x^2 + 3x - 40 = 0 \) has real roots \( x = -8 \) and \( x = 5 \).
2. \( x^2 - 3x - 40 = 0 \) has real roots \( x = -5 \) and \( x = 8 \).
3. \( x^2 - 8x + 112 = 0 \) has complex roots \( x = 4 - 4\sqrt{6}i \) and \( x = 4 + 4\sqrt{6}i \), which are not applicable for real number solutions.

### Next Steps:
- We will now use the real roots of the first two equations to determine the intervals in which the inequality \( x^2 + 3x - 40 \leq 0 \) holds true.
- We need to evaluate the sign of the inequality between the roots where it changes from positive to negative or vice versa.

#### Testing Intervals:
- The roots for \( x^2 + 3x - 40 = 0 \) split the number line into intervals \( x < -8 \), \( -8 < x < 5 \), and \( x > 5 \).
- We will test the inequality \( x^2 + 3x - 40 \leq 0 \) within these intervals using a test point from each interval to check if the inequality is true (i.e., the expression is less than or equal to zero).

Let's test this now.The results of testing the inequality \( x^2 + 3x - 40 \leq 0 \) at the selected points are as follows:

- At \( x = -9 \) (in the interval \( x < -8 \)): \( x^2 + 3x - 40 = 14 \) which is greater than 0, thus the inequality does not hold.
- At \( x = 0 \) (in the interval \( -8 < x < 5 \)): \( x^2 + 3x - 40 = -40 \) which is less than 0, thus the inequality holds.
- At \( x = 6 \) (in the interval \( x > 5 \)): \( x^2 + 3x - 40 = 14 \) which is greater than 0, thus the inequality does not hold.

### Conclusion:
The inequality \( x^2 + 3x - 40 \leq 0 \) is true within the interval \( -8 < x < 5 \). This is the only section of the number line where the quadratic expression remains negative or zero, based on the roots and the test points provided.

Would you like any further details or explanations on this topic? Here are some additional questions you might consider:

1. How do we determine the exact points where the inequality changes from true to false?
2. What are the implications of having complex roots in the context of inequalities?
3. Can we graph the expression \( x^2 + 3x - 40 \) to visually confirm our interval analysis?
4. How does the quadratic formula apply to solving these types of equations?
5. What steps should be taken if we need to solve a similar problem with different coefficients?

**Tip:** When solving inequalities, always consider testing values between the roots, as well as outside the intervals formed by them, to verify where the inequality holds true.

Solve the inequality involving quadratic equations and test within specified intervals.

Learn how to solve quadratic inequalities using the quadratic formula and test values within intervals to check for validity. The solution covers concepts in Algebra, including solving quadratic equations and performing sign analysis for inequalities. Suitable for students in Grades 9-12.

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