Let's analyze the trinomial [tex]\( 9x^2 + 12x + 4 \)[/tex] and determine which one of the given binomials is its factor.
To factorize the trinomial, we look at the provided binomials:
1. [tex]\( 3x + 2 \)[/tex]
2. [tex]\( 3x - 2 \)[/tex]
3. [tex]\( 3x + 1 \)[/tex]
4. [tex]\( 3x - 1 \)[/tex]
### Step-by-Step Verification
#### Option A: [tex]\( 3x + 2 \)[/tex]
Let's test if [tex]\( 3x + 2 \)[/tex] is a factor of [tex]\( 9x^2 + 12x + 4 \)[/tex]:
First, we perform polynomial division of [tex]\( 9x^2 + 12x + 4 \)[/tex] by [tex]\( 3x + 2 \)[/tex]:
1. Divide the leading term [tex]\( 9x^2 \)[/tex] by [tex]\( 3x \)[/tex] to get the first term of the quotient: [tex]\( 3x \)[/tex].
2. Multiply [tex]\( 3x \)[/tex] by [tex]\( 3x + 2 \)[/tex]: [tex]\( (3x)(3x + 2) = 9x^2 + 6x \)[/tex].
3. Subtract this result from the original trinomial: [tex]\( (9x^2 + 12x + 4) - (9x^2 + 6x) = 6x + 4 \)[/tex].
4. Divide the new leading term [tex]\( 6x \)[/tex] by [tex]\( 3x \)[/tex] to get the next term of the quotient: [tex]\( 2 \)[/tex].
5. Multiply [tex]\( 2 \)[/tex] by [tex]\( 3x + 2 \)[/tex]: [tex]\( (2)(3x + 2) = 6x + 4 \)[/tex].
6. Subtract this result from [tex]\( 6x + 4 \)[/tex]: [tex]\( (6x + 4) - (6x + 4) = 0 \)[/tex].
Since the remainder is 0, [tex]\( 3x + 2 \)[/tex] is indeed a factor of [tex]\( 9x^2 + 12x + 4 \)[/tex].
### Conclusion
Based on the detailed verification, we find that Option A, [tex]\( 3x + 2 \)[/tex], is a factor of [tex]\( 9x^2 + 12x + 4 \)[/tex].
Answer: A. [tex]\( 3x + 2 \)[/tex]
