To find the product of the polynomials [tex]\((y^2 + 3y + 7)\)[/tex] and [tex]\((8y^2 + y + 1)\)[/tex], we can use polynomial multiplication. Here, we multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.
The polynomials given are:
[tex]\[
(y^2 + 3y + 7) \quad \text{and} \quad (8y^2 + y + 1)
\][/tex]
Following are steps to multiply these two polynomials:
1. Distribute [tex]\(y^2\)[/tex] to each term in the second polynomial:
[tex]\[
y^2 \cdot (8y^2 + y + 1) = 8y^4 + y^3 + y^2
\][/tex]
2. Distribute [tex]\(3y\)[/tex] to each term in the second polynomial:
[tex]\[
3y \cdot (8y^2 + y + 1) = 24y^3 + 3y^2 + 3y
\][/tex]
3. Distribute [tex]\(7\)[/tex] to each term in the second polynomial:
[tex]\[
7 \cdot (8y^2 + y + 1) = 56y^2 + 7y + 7
\][/tex]
Now, add all these results together:
[tex]\[
8y^4 + y^3 + y^2 + 24y^3 + 3y^2 + 3y + 56y^2 + 7y + 7
\][/tex]
Combine like terms:
[tex]\[
8y^4 + (1y^3 + 24y^3) + (1y^2 + 3y^2 + 56y^2) + (3y + 7y) + 7
\][/tex]
Simplify:
[tex]\[
8y^4 + 25y^3 + 60y^2 + 10y + 7
\][/tex]
Therefore, the product of the polynomials [tex]\((y^2 + 3y + 7)\)[/tex] and [tex]\((8y^2 + y + 1)\)[/tex] is:
[tex]\[
\boxed{8y^4 + 25y^3 + 60y^2 + 10y + 7}
\][/tex]
