To find the product of the polynomials [tex]\((4z^2 + 7z - 8)\)[/tex] and [tex]\((-z + 3)\)[/tex], we will multiply them together step-by-step. The standard approach is to use the distributive property (often referred to as the FOIL method for binomials, but here extended to polynomials).
Let's start multiplying each term from the first polynomial [tex]\((4z^2 + 7z - 8)\)[/tex] with each term from the second polynomial [tex]\((-z + 3)\)[/tex]:
1. Multiplying [tex]\(4z^2\)[/tex] by each term of the second polynomial:
[tex]\[
4z^2 \cdot (-z) = -4z^3
\][/tex]
[tex]\[
4z^2 \cdot 3 = 12z^2
\][/tex]
2. Multiplying [tex]\(7z\)[/tex] by each term of the second polynomial:
[tex]\[
7z \cdot (-z) = -7z^2
\][/tex]
[tex]\[
7z \cdot 3 = 21z
\][/tex]
3. Multiplying [tex]\(-8\)[/tex] by each term of the second polynomial:
[tex]\[
-8 \cdot (-z) = 8z
\][/tex]
[tex]\[
-8 \cdot 3 = -24
\][/tex]
Now, we combine all these products together:
[tex]\[
-4z^3 + 12z^2 - 7z^2 + 21z + 8z - 24
\][/tex]
Next, we combine like terms:
[tex]\[
-4z^3 + (12z^2 - 7z^2) + (21z + 8z) - 24
\][/tex]
Simplifying the like terms:
[tex]\[
-4z^3 + 5z^2 + 29z - 24
\][/tex]
So, the product of [tex]\((4z^2 + 7z - 8)\)[/tex] and [tex]\((-z + 3)\)[/tex] is:
[tex]\[
-4z^3 + 5z^2 + 29z - 24
\][/tex]
Therefore, the completed polynomial product is:
[tex]\[
-4z^3 + 5z^2 + 29z - 24
\][/tex]
And the answer for the given problem, filling in the blanks, is:
[tex]\[
-4z^3 + 5z^2 + 29z - 24
\][/tex]
