Sure, let's go through the process of multiplying the two expressions step-by-step to understand how the product is derived.
We are given two expressions: [tex]\(3a + 5\)[/tex] and [tex]\(2a^2 + 4a - 2\)[/tex].
We need to find the product of these two expressions. To do this, we will use the distributive property (also known as the distributive law of multiplication).
Let's distribute each term in the first expression to each term in the second expression:
[tex]\[
(3a + 5) \cdot (2a^2 + 4a - 2)
\][/tex]
This can be broken down into:
[tex]\[
3a \cdot (2a^2 + 4a - 2) + 5 \cdot (2a^2 + 4a - 2)
\][/tex]
Now, distribute each term:
1. Distribute [tex]\(3a\)[/tex]:
[tex]\[
3a \cdot 2a^2 = 6a^3
\][/tex]
[tex]\[
3a \cdot 4a = 12a^2
\][/tex]
[tex]\[
3a \cdot (-2) = -6a
\][/tex]
2. Distribute [tex]\(5\)[/tex]:
[tex]\[
5 \cdot 2a^2 = 10a^2
\][/tex]
[tex]\[
5 \cdot 4a = 20a
\][/tex]
[tex]\[
5 \cdot (-2) = -10
\][/tex]
Now, combine all these terms together:
[tex]\[
6a^3 + 12a^2 - 6a + 10a^2 + 20a - 10
\][/tex]
Next, we combine like terms:
1. Combine [tex]\(a^3\)[/tex] terms:
[tex]\[
6a^3
\][/tex]
2. Combine [tex]\(a^2\)[/tex] terms:
[tex]\[
12a^2 + 10a^2 = 22a^2
\][/tex]
3. Combine [tex]\(a\)[/tex] terms:
[tex]\[
-6a + 20a = 14a
\][/tex]
4. Constant terms:
[tex]\[
-10
\][/tex]
Putting it all together, we get:
[tex]\[
6a^3 + 22a^2 + 14a - 10
\][/tex]
So, the product of [tex]\(3a + 5\)[/tex] and [tex]\(2a^2 + 4a - 2\)[/tex] is:
[tex]\[
6a^3 + 22a^2 + 14a - 10
\][/tex]
Thus, the correct answer is:
[tex]\[
6a^3 + 22a^2 + 14a - 10
\][/tex]
So, the answer is the first option:
[tex]\[
6a^3 + 22a^2 + 14a - 10
\][/tex]
