To factor the expression [tex]\( -9x^3 - 12x^2 - 4x \)[/tex], let's start by identifying common factors and then further factorizing any remaining terms.
1. Identify the Greatest Common Factor (GCF):
The expression is [tex]\( -9x^3 - 12x^2 - 4x \)[/tex]. We notice that each term has an [tex]\( x \)[/tex] in it, and we can also factor out -1 from all the terms. So, the GCF is [tex]\( -x \)[/tex].
Factor [tex]\( -x \)[/tex] out of each term:
[tex]\[
-x (9x^2 + 12x + 4)
\][/tex]
2. Factor the quadratic expression:
Now we need to factor the quadratic expression [tex]\( 9x^2 + 12x + 4 \)[/tex].
To factor this, let's look at the quadratic part a bit:
[tex]\[
9x^2 + 12x + 4
\][/tex]
This quadratic expression can be written in the form [tex]\( (3x + 2)(3x + 2) \)[/tex] because:
[tex]\[
(3x + 2)^2 = 9x^2 + 6x + 6x + 4 = 9x^2 + 12x + 4
\][/tex]
Therefore, the quadratic expression factors to:
[tex]\[
9x^2 + 12x + 4 = (3x + 2)^2
\][/tex]
3. Combine the factors:
Include the GCF that we factored out in the first step:
[tex]\[
-x (3x + 2)^2
\][/tex]
So, the factored form of the expression [tex]\( -9x^3 - 12x^2 - 4x \)[/tex] is:
[tex]\[
-x(3x + 2)^2
\][/tex]
Based on this solution, the correct answer is:
[tex]\[
\boxed{-x(3x + 2)^2}
\][/tex]
Thus, the correct answer is:
A. [tex]\(-x(3x + 2)^2\)[/tex]
