To find the product of the expressions [tex]\((3a + 2)(4a^2 - 2a + 9)\)[/tex], we will perform polynomial multiplication, specifically the distributive property (also known as the FOIL method for binomials).
Let’s go through the multiplication step by step:
### Step 1: Distribute [tex]\(3a\)[/tex] to each term in [tex]\(4a^2 - 2a + 9\)[/tex]:
[tex]\[
3a \cdot 4a^2 = 12a^3
\][/tex]
[tex]\[
3a \cdot (-2a) = -6a^2
\][/tex]
[tex]\[
3a \cdot 9 = 27a
\][/tex]
So, distributing [tex]\(3a\)[/tex] gives the partial result:
[tex]\[
12a^3 - 6a^2 + 27a
\][/tex]
### Step 2: Distribute [tex]\(2\)[/tex] to each term in [tex]\(4a^2 - 2a + 9\)[/tex]:
[tex]\[
2 \cdot 4a^2 = 8a^2
\][/tex]
[tex]\[
2 \cdot (-2a) = -4a
\][/tex]
[tex]\[
2 \cdot 9 = 18
\][/tex]
So, distributing [tex]\(2\)[/tex] gives another partial result:
[tex]\[
8a^2 - 4a + 18
\][/tex]
### Step 3: Combine all the partial results:
Now, add the results from both distributions together:
[tex]\[
(12a^3 - 6a^2 + 27a) + (8a^2 - 4a + 18)
\][/tex]
Combine the like terms:
[tex]\[
12a^3 + (-6a^2 + 8a^2) + (27a - 4a) + 18
\][/tex]
Simplify the coefficients:
[tex]\[
12a^3 + 2a^2 + 23a + 18
\][/tex]
The correct product of [tex]\((3a + 2)(4a^2 - 2a + 9)\)[/tex] is:
[tex]\[
12a^3 + 2a^2 + 23a + 18
\][/tex]
So, the correct answer is:
[tex]\[
12a^3 + 2a^2 + 23a + 18
\][/tex]
