Solve the polynomial inequality: 
−
4
(
x
+
11
)
(
x
+
5
)
(
x
−
4
)
≥
0


Give your answer in interval notation. Enter DNE if there is no solution.

We are tasked with solving the polynomial inequality:

\[
-4(x + 11)(x + 5)(x - 4) \geq 0
\]

### Step 1: Identify the critical points
The inequality involves a product of three factors: \( (x + 11), (x + 5), \) and \( (x - 4) \). To solve this, we first need to find the critical points, which occur where any factor equals zero. These critical points are the solutions to:

\[
(x + 11) = 0 \quad \Rightarrow \quad x = -11
\]
\[
(x + 5) = 0 \quad \Rightarrow \quad x = -5
\]
\[
(x - 4) = 0 \quad \Rightarrow \quad x = 4
\]

So the critical points are \( x = -11, x = -5, \) and \( x = 4 \).

### Step 2: Determine the sign of the expression in different intervals
The critical points divide the real line into four intervals:
1. \( (-\infty, -11) \)
2. \( (-11, -5) \)
3. \( (-5, 4) \)
4. \( (4, \infty) \)

We now test the sign of the expression \( -4(x + 11)(x + 5)(x - 4) \) in each interval by selecting a test point from each interval.

#### Interval 1: \( (-\infty, -11) \)
Choose \( x = -12 \):
\[
-4(-12 + 11)(-12 + 5)(-12 - 4) = -4(-1)(-7)(-16) = -4(-1)(112) = 448 > 0
\]
So, the expression is positive on \( (-\infty, -11) \).

#### Interval 2: \( (-11, -5) \)
Choose \( x = -10 \):
\[
-4(-10 + 11)(-10 + 5)(-10 - 4) = -4(1)(-5)(-14) = -4(1)(70) = -280 < 0
\]
So, the expression is negative on \( (-11, -5) \).

#### Interval 3: \( (-5, 4) \)
Choose \( x = 0 \):
\[
-4(0 + 11)(0 + 5)(0 - 4) = -4(11)(5)(-4) = -4(11)(-20) = 880 > 0
\]
So, the expression is positive on \( (-5, 4) \).

#### Interval 4: \( (4, \infty) \)
Choose \( x = 5 \):
\[
-4(5 + 11)(5 + 5)(5 - 4) = -4(16)(10)(1) = -640 < 0
\]
So, the expression is negative on \( (4, \infty) \).

### Step 3: Handle the inequality sign and critical points
Since the inequality is \( \geq 0 \), we are interested in intervals where the expression is non-negative (positive or zero). From our analysis, the expression is non-negative on:
- \( (-\infty, -11] \)
- \( [-5, 4] \)

The critical points \( x = -11 \), \( x = -5 \), and \( x = 4 \) are included because the expression equals zero at these points.

### Final answer:
The solution in interval notation is:

\[
(-\infty, -11] \cup [-5, 4]
\]

Would you like further clarification on any steps?

---

Here are 5 related questions to deepen understanding:
1. How would the solution change if the inequality was strict \( > 0 \)?
2. What happens if the polynomial inequality involves higher-degree terms?
3. How would the interval notation change if we excluded critical points where the expression equals zero?
4. How do we determine if a factor contributes positively or negatively to the product in each interval?
5. How does the leading coefficient (here \( -4 \)) affect the direction of inequality?

**Tip**: Always check the sign of the polynomial in each interval defined by the critical points, as this can change the direction of inequality

Solve the polynomial inequality: −4(x + 11)(x + 5)(x − 4) ≥ 0. Give your answer in interval notation. Enter DNE if there is no solution.

Learn how to solve the polynomial inequality −4(x + 11)(x + 5)(x − 4) ≥ 0 with a detailed step-by-step solution. This problem involves identifying critical points, using sign analysis, and writing the final answer in interval notation. Suitable for high school students in Grades 10-12.

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