Solve the following inequality.

x^3+9x^2-108 less or equal to 0.
Then write your solution in interval notation. I am sooo confused and do not have any idea how to do this. Can someone please show me step by step please???

Thanks in advance. Someone who is confused.

This polynomial graphing might help sometimes:

http://mathportal.org/calculators/polynomials-solvers/polynomial-graphing-calculator.php

zeros at x = -6 (double zero) and x = +3
so we really have
(x+6)(x+6)(x-3)

It is negative for large negative x
it comes up and bounces off the x axis at x=-6, dropping down negative again
it does not actually come up and cross the x axis until x = 3
from then on it is positive,
therefore negative for
x</= +3 except for touching zero at x = -6

To solve the inequality x^3 + 9x^2 - 108 ≤ 0, follow these steps:

Step 1: Factorize the expression
Start by factoring out the greatest common factor (GCF). In this case, the GCF is 3:
3(x^3 + 3x^2 - 36) ≤ 0

Now, focus on solving the quadratic expression in parentheses by attempting to factorize it further:
3(x^3 + 3x^2 - 36) ≤ 0

The quadratic expression can be written as:
3(x - 3)(x + 6) ≤ 0

Step 2: Find the critical points
To determine where the expression changes sign, we need to find the critical points. The critical points occur where the expression equals zero. In this case, the critical points are:
x - 3 = 0 ==> x = 3
x + 6 = 0 ==> x = -6

Step 3: Create a number line
Draw a number line and place the critical points on it:
... -6 ... 0 ... 3 ...

Step 4: Test intervals
Choose a value from each interval created by the critical points and substitute it into the original inequality.
For the interval (-∞, -6), we can choose x = -7 as a test value:
3(-7 - 3)(-7 + 6) ≤ 0
3(-10)(-1) ≤ 0
30 ≤ 0 (This is false)

For the interval (-6, 0), we can choose x = -1 as a test value:
3(-1 - 3)(-1 + 6) ≤ 0
3(-4)(5) ≤ 0
-60 ≤ 0 (This is true)

For the interval (0, 3), we can choose x = 1 as a test value:
3(1 - 3)(1 + 6) ≤ 0
3(-2)(7) ≤ 0
-42 ≤ 0 (This is true)

For the interval (3, ∞), we can choose x = 4 as a test value:
3(4 - 3)(4 + 6) ≤ 0
3(1)(10) ≤ 0
30 ≤ 0 (This is false)

Step 5: Determine the solution intervals
Now, analyze the results of the test values on the number line. Mark the intervals where the inequality is true or false.

... -6 ... 0 ... 3 ...
- + -

From the number line, we see that the expression is true for the interval (-6, 0) and (0, 3).

Step 6: Write the solution in interval notation
Using interval notation, the solution to the inequality x^3 + 9x^2 - 108 ≤ 0 is:
(-6, 0] U [0, 3]

This means that x can take on any value within the interval (-6, 0] and the interval [0, 3].