Let's delve into the given polynomial multiplication step-by-step to deduce the correct product:
We have two polynomials:
[tex]\[ (2y - 3) \][/tex]
and
[tex]\[ (3y^2 + 4y - 5) \][/tex]
We need to find the product of these two polynomials.
Step-by-Step Calculation:
1. Distribute each term of the first polynomial with each term of the second polynomial:
2. Multiply [tex]\(2y\)[/tex] with each term in [tex]\(3y^2 + 4y - 5\)[/tex]:
[tex]\[
2y \cdot 3y^2 = 6y^3
\][/tex]
[tex]\[
2y \cdot 4y = 8y^2
\][/tex]
[tex]\[
2y \cdot (-5) = -10y
\][/tex]
3. Multiply [tex]\(-3\)[/tex] with each term in [tex]\(3y^2 + 4y - 5\)[/tex]:
[tex]\[
-3 \cdot 3y^2 = -9y^2
\][/tex]
[tex]\[
-3 \cdot 4y = -12y
\][/tex]
[tex]\[
-3 \cdot (-5) = 15
\][/tex]
Now, add all these individual products together:
[tex]\[
6y^3 + 8y^2 - 10y - 9y^2 - 12y + 15
\][/tex]
4. Combine like terms:
[tex]\[
6y^3 + (8y^2 - 9y^2) + (-10y - 12y) + 15
\][/tex]
[tex]\[
6y^3 - y^2 - 22y + 15
\][/tex]
Therefore, the polynomial resulting from multiplying [tex]\((2y - 3)\)[/tex] and [tex]\((3y^2 + 4y - 5)\)[/tex] is:
[tex]\[ \boxed{6y^3 - y^2 - 22y + 15} \][/tex]
Looking at the given options, the correct match for this polynomial is:
[tex]\[ 6y^3 - y^2 - 22y + 15 \][/tex]
So, the correct polynomial product is:
[tex]\[ \boxed{6y^3 - y^2 - 22y + 15} \][/tex]
from the given answer options list, which corresponds to none of the directly given choices because there's a typo in the provided answers.
