Certainly! Let's find the product of the two given polynomials step by step.
We need to find the product of the expressions: \((y^2 + 3y + 7)\) and \((8y^2 + y + 1)\).
Step 1: Distribute each term in the first polynomial by each term in the second polynomial.
First, let's distribute \(y^2\) across all terms in the second polynomial:
[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]
Next, let's distribute \(3y\) across all terms in the second polynomial:
[tex]\[ 3y \cdot 8y^2 = 24y^3 \][/tex]
[tex]\[ 3y \cdot y = 3y^2 \][/tex]
[tex]\[ 3y \cdot 1 = 3y \][/tex]
Finally, let's distribute \(7\) across all terms in the second polynomial:
[tex]\[ 7 \cdot 8y^2 = 56y^2 \][/tex]
[tex]\[ 7 \cdot y = 7y \][/tex]
[tex]\[ 7 \cdot 1 = 7 \][/tex]
Step 2: Combine all these products together:
[tex]\[ 8y^4 + y^3 + y^2 + 24y^3 + 3y^2 + 3y + 56y^2 + 7y + 7 \][/tex]
Step 3: Group like terms:
[tex]\[ 8y^4 + (y^3 + 24y^3) + (y^2 + 3y^2 + 56y^2) + (3y + 7y) + 7 \][/tex]
Step 4: Simplify these groups:
[tex]\[ 8y^4 + 25y^3 + 60y^2 + 10y + 7 \][/tex]
So, the product of the two polynomials \((y^2 + 3y + 7)\) and \((8y^2 + y + 1)\) is:
[tex]\[ 8y^4 + 25y^3 + 60y^2 + 10y + 7 \][/tex]
Therefore, the correct answer is:
[tex]\[ 8y^4 + 25y^3 + 60y^2 + 10y + 7 \][/tex]
