Sure, let's factorize the polynomial [tex]\(12x^2 + 36x + 27\)[/tex] step by step to find the correct option among the given choices.
### Step 1: Identify Common Factors
First we look for common factors in the polynomial [tex]\(12x^2 + 36x + 27\)[/tex].
The greatest common divisor (GCD) of the coefficients 12, 36, and 27 is 3. We can factor out this common factor:
[tex]\[
12x^2 + 36x + 27 = 3(4x^2 + 12x + 9)
\][/tex]
### Step 2: Factor the Quadratic Inside
Next, we need to factor the quadratic expression [tex]\(4x^2 + 12x + 9\)[/tex].
To factor this trinomial, we look for two numbers that multiply to [tex]\(4 \times 9 = 36\)[/tex] and add to 12. These numbers are 6 and 6. Thus, we can rewrite the quadratic:
[tex]\[
4x^2 + 12x + 9 = 4x^2 + 6x + 6x + 9
\][/tex]
### Step 3: Factor by Grouping
Now we can factor by grouping:
[tex]\[
4x^2 + 6x + 6x + 9 = (4x^2 + 6x) + (6x + 9)
\][/tex]
Factor out the common terms in each group:
[tex]\[
= 2x(2x + 3) + 3(2x + 3)
\][/tex]
Notice that [tex]\((2x + 3)\)[/tex] is a common factor:
[tex]\[
= (2x + 3)(2x + 3)
\][/tex]
This can be written as:
[tex]\[
= (2x + 3)^2
\][/tex]
### Step 4: Incorporate the Common Factor
Recall we factored out a 3 initially. Now we combine it back:
[tex]\[
12x^2 + 36x + 27 = 3(4x^2 + 12x + 9) = 3(2x + 3)^2
\][/tex]
### Final Answer
Now we compare our result with the given options:
A. [tex]\((x + 3)(12x + 9)\)[/tex]
B. [tex]\(3(2x + 3)^2\)[/tex]
C. [tex]\(6(x + 2)^2\)[/tex]
D. [tex]\((6x + 3)(2x + 9)\)[/tex]
The correct factorization is:
[tex]\[
12 x^2 + 36 x + 27 = 3(2 x + 3)^2
\][/tex]
This matches option B.
### Conclusion
The complete factorization of [tex]\(12 x^2+36 x+27\)[/tex] is given by:
[tex]\[
\boxed{B. \, 3(2 x + 3)^2}
\][/tex]
