Certainly! Let's find the product of the two polynomials:
[tex]\[
(4y - 3) \left(2y^2 + 3y - 5\right)
\][/tex]
To do this, we'll distribute each term in the first polynomial to each term in the second polynomial and then combine like terms.
1. First, distribute [tex]\(4y\)[/tex]:
[tex]\[
4y \cdot 2y^2 = 8y^3
\][/tex]
[tex]\[
4y \cdot 3y = 12y^2
\][/tex]
[tex]\[
4y \cdot (-5) = -20y
\][/tex]
2. Next, distribute [tex]\(-3\)[/tex]:
[tex]\[
-3 \cdot 2y^2 = -6y^2
\][/tex]
[tex]\[
-3 \cdot 3y = -9y
\][/tex]
[tex]\[
-3 \cdot (-5) = 15
\][/tex]
3. Now combine all these results:
[tex]\[
8y^3 + 12y^2 - 20y - 6y^2 - 9y + 15
\][/tex]
4. Combine like terms:
[tex]\[
8y^3 + (12y^2 - 6y^2) + (-20y - 9y) + 15
\][/tex]
[tex]\[
8y^3 + 6y^2 - 29y + 15
\][/tex]
So, the product is:
[tex]\[
8y^3 + 6y^2 - 29y + 15
\][/tex]
From the given choices, the correct one is:
[tex]\[
8 y^3 + 6 y^2 - 29 y + 15
\][/tex]
