https://file.aiquickdraw.com/mathbot/file/1726152734571p3opb2zk.jpg

The inequality provided in the image is:

\[
x^2 + 15x + 44 \geq 0
\]

Let's solve this step by step:

### 1. Solve the quadratic equation:
First, find the roots of the quadratic equation:

\[
x^2 + 15x + 44 = 0
\]

We will use the quadratic formula:

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

Here, \( a = 1 \), \( b = 15 \), and \( c = 44 \).

### 2. Calculate the discriminant:

\[
\Delta = b^2 - 4ac = 15^2 - 4(1)(44) = 225 - 176 = 49
\]

### 3. Find the roots:

\[
x = \frac{-15 \pm \sqrt{49}}{2(1)} = \frac{-15 \pm 7}{2}
\]

This gives two solutions:

\[
x_1 = \frac{-15 + 7}{2} = \frac{-8}{2} = -4
\]
\[
x_2 = \frac{-15 - 7}{2} = \frac{-22}{2} = -11
\]

### 4. Analyze the inequality:

Since the discriminant is positive, the quadratic expression factors into:

\[
(x + 4)(x + 11) \geq 0
\]

To solve this inequality, we look at the sign of the product \( (x + 4)(x + 11) \) based on the critical points \( x = -4 \) and \( x = -11 \).

The sign table for the intervals is:

- For \( x < -11 \), both factors \( (x + 4) \) and \( (x + 11) \) are negative, so the product is positive.
- For \( -11 < x < -4 \), one factor is negative and the other is positive, so the product is negative.
- For \( x > -4 \), both factors are positive, so the product is positive.

Thus, the inequality \( (x + 4)(x + 11) \geq 0 \) holds for:

\[
x \leq -11 \quad \text{or} \quad x \geq -4
\]

### Final answer:

\[
x \leq -11 \quad \text{or} \quad x \geq -4
\]

Would you like further details or have any questions?

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**Related Questions:**

1. How do you solve quadratic inequalities in general?
2. What does the discriminant tell us about the roots of a quadratic equation?
3. What are critical points in inequality solutions?
4. How does the sign chart method work for quadratic inequalities?
5. Can we apply this method to non-quadratic polynomials?

**Tip:** Always check the sign of the product in each interval between critical points when solving polynomial inequalities.

Solve the inequality x^2 + 15x + 44 ≥ 0.

Learn how to solve the quadratic inequality x^2 + 15x + 44 ≥ 0 using the quadratic formula and sign chart method. This problem covers key algebraic concepts like factoring and analyzing quadratic inequalities. Suitable for high school students (Grades 9-12).

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