https://file.aiquickdraw.com/mathbot/file/1727785046695xu7q5v38.jpg

The inequality given in the image is:

\[
-2x^2 + 9x + 5 > 0
\]

We need to solve this quadratic inequality and find the intervals where it holds true.

### Step 1: Solve the corresponding quadratic equation
To solve the inequality, we first solve the equation:

\[
-2x^2 + 9x + 5 = 0
\]

Using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \(a = -2\), \(b = 9\), and \(c = 5\), we calculate:

\[
x = \frac{-9 \pm \sqrt{9^2 - 4(-2)(5)}}{2(-2)} = \frac{-9 \pm \sqrt{81 + 40}}{-4} = \frac{-9 \pm \sqrt{121}}{-4} = \frac{-9 \pm 11}{-4}
\]

This gives two solutions:

\[
x_1 = \frac{-9 + 11}{-4} = \frac{2}{-4} = -\frac{1}{2}
\]
\[
x_2 = \frac{-9 - 11}{-4} = \frac{-20}{-4} = 5
\]

Thus, the solutions to the quadratic equation are \(x = -\frac{1}{2}\) and \(x = 5\).

### Step 2: Analyze the inequality
Next, we need to determine where the quadratic expression is greater than zero. To do this, we analyze the sign of the expression in the intervals determined by the roots \(x = -\frac{1}{2}\) and \(x = 5\).

The intervals are:
- \( (-\infty, -\frac{1}{2}) \)
- \( \left( -\frac{1}{2}, 5 \right) \)
- \( (5, \infty) \)

Now, test the inequality \( -2x^2 + 9x + 5 > 0 \) in each interval by picking test points.

1. **For \( x \in (-\infty, -\frac{1}{2}) \)** (e.g., \(x = -1\)):
   \[
   -2(-1)^2 + 9(-1) + 5 = -2 - 9 + 5 = -6
   \]
   This is negative, so the inequality is not true in this interval.

2. **For \( x \in \left( -\frac{1}{2}, 5 \right) \)** (e.g., \(x = 0\)):
   \[
   -2(0)^2 + 9(0) + 5 = 5
   \]
   This is positive, so the inequality is true in this interval.

3. **For \( x \in (5, \infty) \)** (e.g., \(x = 6\)):
   \[
   -2(6)^2 + 9(6) + 5 = -72 + 54 + 5 = -13
   \]
   This is negative, so the inequality is not true in this interval.

### Step 3: Conclusion
The inequality \( -2x^2 + 9x + 5 > 0 \) holds for \( x \in \left( -\frac{1}{2}, 5 \right) \).

Thus, the solution is:

\[
L = \left( -\frac{1}{2}, 5 \right)
\]

### Formatting of the answer
Based on the instructions on the image, the solution should be entered using interval notation. The correct interval to input would be:

\[
L = (-0.5, 5)
\]

Would you like more details on any part of this solution?

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Here are 5 questions to expand on this topic:
1. How does the sign of the leading coefficient of a quadratic affect the direction of the parabola?
2. What other methods can be used to solve quadratic inequalities?
3. How would you approach solving a quadratic inequality where the quadratic cannot be factored?
4. Why do we test points in intervals between the roots when solving inequalities?
5. How can you graphically represent the solution to the quadratic inequality?

**Tip:** When solving quadratic inequalities, always start by solving the corresponding quadratic equation to identify the critical points where the inequality might change.

Learn how to solve the quadratic inequality -2x^2 + 9x + 5 > 0 using the quadratic formula and sign analysis. The problem involves solving the inequality by finding critical points and testing intervals. Suitable for students in Grades 10-12.

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