To solve the inequality \( 2x^2 + 9x + 4 \geq 0 \), we will go through the following steps:

1. **Find the roots of the quadratic equation:** First, solve the related quadratic equation \( 2x^2 + 9x + 4 = 0 \) using the quadratic formula:
   
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
   \]
   
   where \( a = 2 \), \( b = 9 \), and \( c = 4 \). Plugging in the values:

   \[
   x = \frac{-9 \pm \sqrt{9^2 - 4 \times 2 \times 4}}{2 \times 2} = \frac{-9 \pm \sqrt{81 - 32}}{4} = \frac{-9 \pm \sqrt{49}}{4}.
   \]
   
   Since \( \sqrt{49} = 7 \), the roots are:

   \[
   x = \frac{-9 + 7}{4} = \frac{-2}{4} = -\frac{1}{2}, \quad x = \frac{-9 - 7}{4} = \frac{-16}{4} = -4.
   \]

2. **Determine intervals:** The roots divide the number line into three intervals:
   
   \[
   (-\infty, -4), \quad (-4, -\frac{1}{2}), \quad \text{and} \quad \left(-\frac{1}{2}, \infty\right).
   \]

3. **Test each interval:** Since the coefficient of \( x^2 \) is positive (\( 2 > 0 \)), the parabola opens upwards. Therefore, the quadratic expression \( 2x^2 + 9x + 4 \) is positive outside the interval defined by its roots and negative inside. This means:

   - The quadratic is **positive** for \( x \in (-\infty, -4] \) and \( x \in \left[-\frac{1}{2}, \infty\right) \).
   - The quadratic is **negative** for \( x \in (-4, -\frac{1}{2}) \).

4. **Write the solution:** Since we are looking for where \( 2x^2 + 9x + 4 \geq 0 \), the solution includes the intervals where the expression is positive or zero:

   \[
   x \in (-\infty, -4] \cup \left[-\frac{1}{2}, \infty\right).
   \]

Thus, the solution to the inequality is:

\[
x \in (-\infty, -4] \cup \left[-\frac{1}{2}, \infty\right).
\]

Would you like further details or have any questions?

Here are some related questions you might find helpful:
1. How do you find the vertex of a quadratic equation?
2. How can you test intervals for inequalities involving polynomials?
3. What does the discriminant tell you about the roots of a quadratic equation?
4. How do you solve quadratic inequalities graphically?
5. How do you determine whether a quadratic opens upwards or downwards?

**Tip:** Always remember that the sign of the coefficient of \( x^2 \) in a quadratic equation determines whether the parabola opens upwards (positive) or downwards (negative). This helps in quickly identifying where the inequality might hold true.

Learn how to solve the quadratic inequality 2x^2 + 9x + 4 ≥ 0 with a step-by-step guide. This solution covers quadratic equations, the quadratic formula, and interval testing. Ideal for students in Grades 9-11 studying algebra.

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