Certainly! Let's find the product of the expression:
[tex]\[
3x^5 \left(2x^2 + 4x + 1\right)
\][/tex]
We need to distribute [tex]\(3x^5\)[/tex] to each term within the parentheses:
1. Distribute [tex]\(3x^5\)[/tex] to [tex]\(2x^2\)[/tex]:
[tex]\[
3x^5 \cdot 2x^2 = 6x^{5+2} = 6x^7
\][/tex]
2. Distribute [tex]\(3x^5\)[/tex] to [tex]\(4x\)[/tex]:
[tex]\[
3x^5 \cdot 4x = 12x^{5+1} = 12x^6
\][/tex]
3. Distribute [tex]\(3x^5\)[/tex] to [tex]\(1\)[/tex]:
[tex]\[
3x^5 \cdot 1 = 3x^5
\][/tex]
Now, combine all these results together:
[tex]\[
6x^7 + 12x^6 + 3x^5
\][/tex]
Thus, the expanded expression is:
[tex]\[
6x^7 + 12x^6 + 3x^5
\][/tex]
So the correct answer is:
[tex]\[
6x^7 + 12x^6 + 3x^5
\][/tex]
This corresponds to the option:
[tex]\[
\boxed{6x^7 + 12x^6 + 3x^5}
\][/tex]
