solve the polynomial inequality and graph the solution set on a real number line. set in interval notation.

x^3+2x^2-4x-8>0

I have been staring at this problem for the last 15 minutes because I just don't understand it, somebody please help me?

x^3+2x^2-4x-8 > 0

(x+2)^2 (x-2) > 0

Now, you know that the graph touches the x-axis at x = -2 and cross it at x=2

Since y<0 for large x<0, that means that y<0 for x < -2, and stays <0 up to x=2, where it crosses the x-axis.

So, the solution set is x>2

To solve the polynomial inequality x^3 + 2x^2 - 4x - 8 > 0, we need to find the values of x that make the inequality true.

Step 1: Factor the given polynomial if possible. In this case, we cannot easily factor the polynomial.

Step 2: Find the critical points by setting the polynomial equal to zero and solving for x.

Setting the polynomial equal to zero: x^3 + 2x^2 - 4x - 8 = 0

Unfortunately, there are no obvious values that satisfy this equation. Therefore, we need to use other methods to find the critical points.

Step 3: Use a graphing calculator or graphing software to plot the polynomial.

Using a graphing calculator, plot the function y = x^3 + 2x^2 - 4x - 8. The points where the graph intersects or crosses the x-axis are the critical points.

Step 4: Determine the sign of the polynomial in each interval between the critical points.

To determine the sign, choose any x-value within each interval and substitute it into the polynomial. If the resulting value is positive, then the polynomial is positive in that interval. If the resulting value is negative, then the polynomial is negative in that interval.

Step 5: Write the solution set.

Based on the sign of the polynomial in each interval, we can determine the solution set.

Now, let's go through the steps:

Step 1: We cannot factor the given polynomial easily.

Step 2: Let's use a graphing calculator to plot the function y = x^3 + 2x^2 - 4x - 8. By analyzing the graph, we can find the critical points:

Critical point 1: x ≈ -3.64
Critical point 2: x ≈ -1.55

Step 3: Now, we'll determine the sign of the polynomial in each interval between the critical points:

Interval 1: (-∞, -3.64)
Choose any x-value in this interval, for example, -5:
Substituting -5 into the polynomial, we get: (-5)^3 + 2(-5)^2 - 4(-5) - 8 = -50
Since the result is negative, the polynomial is negative in this interval.

Interval 2: (-3.64, -1.55)
Choose any x-value in this interval, for example, -2:
Substituting -2 into the polynomial, we get: (-2)^3 + 2(-2)^2 - 4(-2) - 8 = 0
Since the result is zero, the polynomial is neither positive nor negative in this interval.

Interval 3: (-1.55, +∞)
Choose any x-value in this interval, for example, 0:
Substituting 0 into the polynomial, we get: 0^3 + 2(0)^2 - 4(0) - 8 = -8
Since the result is negative, the polynomial is negative in this interval.

Step 4: Based on the sign of the polynomial in each interval, we can write the solution set:

Solution set: (-∞, -3.64) U (-1.55, +∞)

Step 5: Write the solution set in interval notation:

Interval notation: (-∞, -3.64) U (-1.55, +∞)

Finally, the graph of the solution set on a real number line would have an open circle at -3.64 and -1.55, and would be shaded to the left of -3.64 and to the right of -1.55.