Let's find the product of the expressions [tex]\(3a + 5\)[/tex] and [tex]\(2a^2 + 4a - 2\)[/tex] step-by-step.
We'll use the distributive property (also known as the FOIL method for binomials, but extended here):
1. Distribute each term in [tex]\(3a + 5\)[/tex] to every term in [tex]\(2a^2 + 4a - 2\)[/tex]:
[tex]\[
(3a + 5) \cdot (2a^2 + 4a - 2)
\][/tex]
2. Distribute [tex]\(3a\)[/tex] to each term in [tex]\(2a^2 + 4a - 2\)[/tex]:
[tex]\[
3a \cdot 2a^2 + 3a \cdot 4a + 3a \cdot (-2)
\][/tex]
Calculate each term:
[tex]\[
3a \cdot 2a^2 = 6a^3
\][/tex]
[tex]\[
3a \cdot 4a = 12a^2
\][/tex]
[tex]\[
3a \cdot (-2) = -6a
\][/tex]
3. Distribute [tex]\(5\)[/tex] to each term in [tex]\(2a^2 + 4a - 2\)[/tex]:
[tex]\[
5 \cdot 2a^2 + 5 \cdot 4a + 5 \cdot (-2)
\][/tex]
Calculate each term:
[tex]\[
5 \cdot 2a^2 = 10a^2
\][/tex]
[tex]\[
5 \cdot 4a = 20a
\][/tex]
[tex]\[
5 \cdot (-2) = -10
\][/tex]
4. Combine all the terms together:
[tex]\[
6a^3 + 12a^2 - 6a + 10a^2 + 20a - 10
\][/tex]
5. Combine like terms:
[tex]\[
6a^3 + (12a^2 + 10a^2) + (-6a + 20a) - 10
\][/tex]
Simplify each group of like terms:
[tex]\[
6a^3 + 22a^2 + 14a - 10
\][/tex]
Thus, the product of [tex]\(3a + 5\)[/tex] and [tex]\(2a^2 + 4a - 2\)[/tex] is [tex]\(\boxed{6a^3 + 22a^2 + 14a - 10}\)[/tex].
