To find the product [tex]\((y^2 + 3y + 7)(8y^2 + y + 1)\)[/tex], we need to multiply the terms in each polynomial step-by-step and then combine like terms. Here's the detailed breakdown:
1. First, distribute each term of the first polynomial by each term of the second polynomial.
[tex]\[
(y^2 + 3y + 7)(8y^2 + y + 1)
\][/tex]
2. Distribute [tex]\(y^2\)[/tex] across the second polynomial:
[tex]\[
y^2 \cdot 8y^2 + y^2 \cdot y + y^2 \cdot 1 = 8y^4 + y^3 + y^2
\][/tex]
3. Distribute [tex]\(3y\)[/tex] across the second polynomial:
[tex]\[
3y \cdot 8y^2 + 3y \cdot y + 3y \cdot 1 = 24y^3 + 3y^2 + 3y
\][/tex]
4. Distribute [tex]\(7\)[/tex] across the second polynomial:
[tex]\[
7 \cdot 8y^2 + 7 \cdot y + 7 \cdot 1 = 56y^2 + 7y + 7
\][/tex]
5. Now, combine all the products:
[tex]\[
8y^4 + y^3 + y^2 + 24y^3 + 3y^2 + 3y + 56y^2 + 7y + 7
\][/tex]
6. Combine like terms:
[tex]\[
8y^4 + (1y^3 + 24y^3) + (1y^2 + 3y^2 + 56y^2) + (3y + 7y) + 7
\][/tex]
[tex]\[
8y^4 + 25y^3 + 60y^2 + 10y + 7
\][/tex]
So, the product of the polynomials [tex]\((y^2 + 3y + 7)(8y^2 + y + 1)\)[/tex] is:
[tex]\[
8y^4 + 25y^3 + 60y^2 + 10y + 7
\][/tex]
The correct answer is:
[tex]\[
8y^4 + 25y^3 + 60y^2 + 10y + 7
\][/tex]
