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

x^3+x^2+64x+64>0

show work please

To solve the polynomial inequality x^3 + x^2 + 64x + 64 > 0, we need to find the values of x that satisfy the inequality. Here's the step-by-step process:

Step 1: Factor the polynomial if possible.
Unfortunately, the given polynomial cannot be easily factored.

Step 2: Find the critical points or roots.
To solve the inequality, we need to find the values of x where the polynomial is equal to zero. This means we need to solve the equation x^3 + x^2 + 64x + 64 = 0.

Since factoring the equation is challenging, we can use numerical methods or a graphing calculator to find the approximate solutions. Based on calculations, the polynomial has one real root, which is approximately -4.709598.

Step 3: Determine the sign of the polynomial in each region.
To determine the sign of the polynomial in each region, we can choose test points within each interval. Two commonly chosen test points are -10 and 0. By substituting these test points into the polynomial, we can observe the sign changes.

1. For x < -4.709598:
Substituting x = -10, we get:
(-10)^3 + (-10)^2 + 64(-10) + 64
= -1000 + 100 - 640 + 64 = -1476
The sign is negative.

2. For -4.709598 < x < 0:
Substituting x = 0, we get:
(0)^3 + (0)^2 + 64(0) + 64
= 0 + 0 + 0 + 64 = 64
The sign is positive.

3. For x > 0:
Substituting x = 10, we get:
(10)^3 + (10)^2 + 64(10) + 64
= 1000 + 100 + 640 + 64 = 1804
The sign is positive.

Step 4: Express the solution set in interval notation.
Based on the sign analysis, we can now express the solution set in interval notation:

The solution set is (-∞, -4.709598) U (0, +∞).

And that's how you solve the polynomial inequality and express the solution set in interval notation!