What is the product? (y^2+3y+7)(8y^2+y+1)

To multiply two polynomials, multiply every term of the first polynomial by every term of the second polynomial. Then combine like terms.

I like to take the each term of the first polynomial and multiply by every term of the second polynomial.

(y^2 + 3y + 7)(8y^2 + y + 1) =

= y^2 * 8y^2 + y^2 * y + y^2 * 1 + 3y * 8y^2 + 3y * y + 3y * 1 + 7 * 8y^2
 + 7 * y + 7 * 1      < first step of solution

= 8y^4 + y^3 + y^2 + 24y^3 + 3y^2 + 3y + 56y^2 + 7y + 7  <--- notice 9 terms

= 8y^4 + 25y^3 + 60y^2 + 10y + 7

Notes:
1) I usually don't write the first step of my solution. I wrote it here to show you all the multiplications of terms that need to be done.
2) As a check, make sure the number of products of terms equals the product of the numbers of terms of the two polynomials. These polynomials have 3 terms each. 3 * 3 = 9, and sure enough, there are 9 products in the first step of the solution.