To find the product of [tex]\((5r - 4)\)[/tex] and [tex]\((r^2 - 6r + 4)\)[/tex], we need to use the distributive property, also known as the FOIL method, which stands for First, Outer, Inner, Last. This method helps in expanding the product of two binomials or generally in distributing each term of the first polynomial to every term of the second polynomial.
Let's break down the multiplication step by step:
1. Distribute [tex]\(5r\)[/tex] to each term in [tex]\(r^2 - 6r + 4\)[/tex]:
[tex]\[
5r \cdot r^2 = 5r^3
\][/tex]
[tex]\[
5r \cdot (-6r) = -30r^2
\][/tex]
[tex]\[
5r \cdot 4 = 20r
\][/tex]
2. Distribute [tex]\(-4\)[/tex] to each term in [tex]\(r^2 - 6r + 4\)[/tex]:
[tex]\[
-4 \cdot r^2 = -4r^2
\][/tex]
[tex]\[
-4 \cdot (-6r) = 24r
\][/tex]
[tex]\[
-4 \cdot 4 = -16
\][/tex]
3. Combine all the terms:
[tex]\[
5r^3 - 30r^2 + 20r - 4r^2 + 24r - 16
\][/tex]
4. Combine like terms:
[tex]\[
5r^3 + (-30r^2 - 4r^2) + (20r + 24r) - 16
\][/tex]
[tex]\[
5r^3 - 34r^2 + 44r - 16
\][/tex]
So, the product of [tex]\((5r - 4)\)[/tex] and [tex]\((r^2 - 6r + 4)\)[/tex] is:
[tex]\[
5r^3 - 34r^2 + 44r - 16
\][/tex]
Thus, the correct answer is [tex]\(5r^3 - 34r^2 + 44r - 16\)[/tex].
