To find the product of the two given expressions [tex]\(y^2 + 3y + 7\)[/tex] and [tex]\(8y^2 + y + 1\)[/tex], we need to use polynomial multiplication. The detailed, step-by-step solution is as follows:
1. Write down the two polynomials:
[tex]\[
(y^2 + 3y + 7) \quad \text{and} \quad (8y^2 + y + 1)
\][/tex]
2. Multiply each term in the first polynomial by each term in the second polynomial. The terms should be:
[tex]\[
y^2 \cdot 8y^2 = 8y^4
\][/tex]
[tex]\[
y^2 \cdot y = y^3
\][/tex]
[tex]\[
y^2 \cdot 1 = y^2
\][/tex]
[tex]\[
3y \cdot 8y^2 = 24y^3
\][/tex]
[tex]\[
3y \cdot y = 3y^2
\][/tex]
[tex]\[
3y \cdot 1 = 3y
\][/tex]
[tex]\[
7 \cdot 8y^2 = 56y^2
\][/tex]
[tex]\[
7 \cdot y = 7y
\][/tex]
[tex]\[
7 \cdot 1 = 7
\][/tex]
3. Now add together all like terms, grouping them by the power of [tex]\(y\)[/tex]:
[tex]\[
8y^4
\][/tex]
[tex]\[
y^3 + 24y^3 = 25y^3
\][/tex]
[tex]\[
y^2 + 3y^2 + 56y^2 = 60y^2
\][/tex]
[tex]\[
3y + 7y = 10y
\][/tex]
[tex]\[
7
\][/tex]
4. Combine these like terms to form the final product:
[tex]\[
8y^4 + 25y^3 + 60y^2 + 10y + 7
\][/tex]
Thus, the product of the polynomials [tex]\(y^2 + 3y + 7\)[/tex] and [tex]\(8y^2 + y + 1\)[/tex] is:
[tex]\[
8y^4 + 25y^3 + 60y^2 + 10y + 7
\][/tex]
This matches with one of the options provided, specifically:
[tex]\[
8y^4 + 25y^3 + 60y^2 + 10y + 7
\][/tex]
