To find the product of the two polynomials [tex]\(\left(y^2 + 3y + 7\right) \left(8y^2 + y + 1\right)\)[/tex], we'll multiply the two expressions step-by-step and combine like terms.
Step 1: Distribute [tex]\(y^2\)[/tex]:
[tex]\[
y^2 \cdot (8y^2) + y^2 \cdot (y) + y^2 \cdot (1)
= 8y^4 + y^3 + y^2
\][/tex]
Step 2: Distribute [tex]\(3y\)[/tex]:
[tex]\[
3y \cdot (8y^2) + 3y \cdot (y) + 3y \cdot (1)
= 24y^3 + 3y^2 + 3y
\][/tex]
Step 3: Distribute [tex]\(7\)[/tex]:
[tex]\[
7 \cdot (8y^2) + 7 \cdot (y) + 7 \cdot (1)
= 56y^2 + 7y + 7
\][/tex]
Step 4: Combine the results from steps 1, 2, and 3:
[tex]\[
8y^4 + y^3 + y^2 + 24y^3 + 3y^2 + 3y + 56y^2 + 7y + 7
\][/tex]
Step 5: Combine like terms:
- [tex]\(8y^4\)[/tex] (the only [tex]\(y^4\)[/tex] term)
- [tex]\(1y^3 + 24y^3 = 25y^3\)[/tex]
- [tex]\(y^2 + 3y^2 + 56y^2 = 60y^2\)[/tex]
- [tex]\(3y + 7y = 10y\)[/tex]
- [tex]\(+7\)[/tex] (the constant term)
So, the combined polynomial is:
[tex]\[
8y^4 + 25y^3 + 60y^2 + 10y + 7
\][/tex]
Therefore, the correct product is:
[tex]\[
\boxed{4}
\][/tex]
