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The image presents a multiple-choice question that asks, "Which of these inequalities has solution \(-2 < x < 3\)?"

The options are:

1. \( x^2 < x + 6 \)
2. \( x^2 + x < 6 \)
3. \( x^2 + 6 < x \)
4. \( x^2 + x + 6 < 0 \)

Let's analyze these one by one to find which inequality has the solution \(-2 < x < 3\).

### Option 1: \( x^2 < x + 6 \)
Rearrange this inequality: 
\[
x^2 - x - 6 < 0
\]
Factor the quadratic:
\[
(x - 3)(x + 2) < 0
\]
The solution to this inequality is when the expression is negative. By analyzing the critical points, \(x = -2\) and \(x = 3\), the solution is indeed \(-2 < x < 3\).

### Option 2: \( x^2 + x < 6 \)
Rearrange:
\[
x^2 + x - 6 < 0
\]
Factor the quadratic:
\[
(x - 2)(x + 3) < 0
\]
The solution is \(-3 < x < 2\), which is not the given solution.

### Option 3: \( x^2 + 6 < x \)
Rearrange:
\[
x^2 - x + 6 < 0
\]
This quadratic has no real solutions because the discriminant (\(b^2 - 4ac\)) is negative. Hence, this inequality has no solution.

### Option 4: \( x^2 + x + 6 < 0 \)
The quadratic equation \(x^2 + x + 6\) always has a positive value for all real \(x\), so it never becomes negative. Hence, this inequality has no solution.

### Conclusion:
The correct inequality that has the solution \(-2 < x < 3\) is **Option 1: \(x^2 < x + 6\)**.

Would you like further details or explanations? Here are some related questions:

1. How do you solve quadratic inequalities graphically?
2. What is the significance of the discriminant in determining the nature of quadratic inequalities?
3. How can factoring help solve inequalities?
4. What is the relationship between critical points and the solution of inequalities?
5. How can you test a solution interval for inequalities?

**Tip:** When solving quadratic inequalities, always check the sign of the expression in different intervals between the critical points!

Which of these inequalities has solution -2 < x < 3?

Learn how to solve the quadratic inequality that has the solution -2 < x < 3. This problem involves key algebraic concepts such as quadratic inequalities and factoring. Suitable for students in Grades 9-11.

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