To factor the expression [tex]\(-9x^3 - 12x^2 - 4x\)[/tex], follow these steps:
1. Identify the Greatest Common Factor (GCF):
The GCF of the terms [tex]\(-9x^3\)[/tex], [tex]\(-12x^2\)[/tex], and [tex]\(-4x\)[/tex] is [tex]\(-x\)[/tex].
2. Factor out the GCF:
By factoring [tex]\(-x\)[/tex] from each term, we get:
[tex]\[
-x(9x^2 + 12x + 4)
\][/tex]
3. Recognize the quadratic expression:
Now we need to factor the quadratic expression [tex]\(9x^2 + 12x + 4\)[/tex]. The quadratic is of the form [tex]\(ax^2 + bx + c\)[/tex].
4. Factor the quadratic expression:
The quadratic [tex]\(9x^2 + 12x + 4\)[/tex] can be factored into a perfect square trinomial:
[tex]\[
9x^2 + 12x + 4 = (3x + 2)^2
\][/tex]
5. Combine the factored terms:
Substitute [tex]\( (3x + 2)^2 \)[/tex] back into the expression we factored out earlier:
[tex]\[
-x(9x^2 + 12x + 4) = -x(3x + 2)^2
\][/tex]
Therefore, the fully factored form of the expression [tex]\(-9x^3 - 12x^2 - 4x\)[/tex] is [tex]\(-x(3x + 2)^2\)[/tex].
So, the correct answer is:
D. [tex]\(-x(3x+2)^2\)[/tex]
