Let's find the product [tex]\(\left(y^2 + 3y + 7 \right) \left(8y^2 + y + 1 \right)\)[/tex] step by step. We'll distribute each term in the first polynomial to every term in the second polynomial and then combine like terms.
1. Distribute [tex]\(y^2\)[/tex] in the first polynomial:
[tex]\[
y^2 \cdot (8y^2 + y + 1) = 8y^4 + y^3 + y^2
\][/tex]
2. Distribute [tex]\(3y\)[/tex] in the first polynomial:
[tex]\[
3y \cdot (8y^2 + y + 1) = 24y^3 + 3y^2 + 3y
\][/tex]
3. Distribute [tex]\(7\)[/tex] in the first polynomial:
[tex]\[
7 \cdot (8y^2 + y + 1) = 56y^2 + 7y + 7
\][/tex]
Now, we combine all these products:
[tex]\[
(8y^4 + y^3 + y^2) + (24y^3 + 3y^2 + 3y) + (56y^2 + 7y + 7)
\][/tex]
Next, we combine like terms:
- For [tex]\(y^4\)[/tex]:
[tex]\[
8y^4
\][/tex]
- For [tex]\(y^3\)[/tex]:
[tex]\[
y^3 + 24y^3 = 25y^3
\][/tex]
- For [tex]\(y^2\)[/tex]:
[tex]\[
y^2 + 3y^2 + 56y^2 = 60y^2
\][/tex]
- For [tex]\(y\)[/tex]:
[tex]\[
3y + 7y = 10y
\][/tex]
- Constant term:
[tex]\[
7
\][/tex]
So, the final product is:
[tex]\[
8y^4 + 25y^3 + 60y^2 + 10y + 7
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{8y^4 + 25y^3 + 60y^2 + 10y + 7}
\][/tex]
