I have posted this question earlier and had the answer given to me this way. But my teacher needs to know what type of factorization I used and I have tried to figure it out but have NO clue!!

This is the question.
Solve the following inequality. write your answer in interval notation.
x^3+9x^2-108 less than or equal to o

My anser:
(x+6)(x+6)(x-3)
we have double zeros at x= -6
graph comes from way low and bounces back down off the x-axis at x= -6

Here is the rest of the answer:

then dropping down negative again

then it comes back up again and goes positive and crosses the x-axis at x=3
and from then on is positive

x=3
(-00,3)u(0,6)
**Now she says note that the last term does not contain an x. What type of method did I use in this posting. Explain this method. Can someone help me please

Not sure what you're going on about.

What do you mean "type of factorization"?
I used synthetic division to come up with the roots. If you have no clue, then you need to review finding roots of a function.

The roots are where the graph touches or crosses the x-axis.

The question is:

solve f(x) <= 0

The answer is:

f(x) <= 0 when x <= 3.
In interval notation, x is in (-oo,3]

To understand the type of factorization used in solving the inequality, let's break down the process step by step:

Step 1: Write the inequality in the form of an equation:
x^3 + 9x^2 - 108 ≤ 0

Step 2: Factorize the equation:
To factorize the equation, you need to find the roots or zeros of the equation. In this case, you already provided the factorization (x+6)(x+6)(x-3).

Step 3: Find the critical points:
To determine where the function crosses the x-axis (zeros), we set each factor equal to zero and solve for x:
x+6 = 0 --> x = -6
x+6 = 0 --> x = -6
x-3 = 0 --> x = 3

Step 4: Determine the intervals:
Now, you need to determine which intervals represent the solutions to the inequality. To do this, you can use the critical points as guidelines.

Since we have the factorization (x+6)(x+6)(x-3) and the inequality is "less than or equal to zero," it means the function will be negative or zero in the intervals determined by these critical points.

-∞ < -6 : In this interval, the signs of the factors are (-)(-)(-) = -, which means the function is negative or zero.
-6 < 3 : In this interval, the signs of the factors are (+)(-)(-) = +, which means the function is positive.
3 < ∞ : In this interval, the signs of the factors are (+)(+)(-) = -, which means the function is negative or zero.

Step 5: Write the solution in interval notation:
Based on the signs of the factors, the solutions to the inequality are the intervals where the function is negative or zero, which are (-∞, -6] and [3, ∞).

Therefore, the solution to the inequality x^3 + 9x^2 - 108 ≤ 0, written in interval notation, is (-∞, -6] ∪ [3, ∞).