To find the product of the given expressions [tex]\((5r + 2)(3r - 4)\)[/tex], let's proceed with the steps involved in polynomial multiplication. We will apply the distributive property (also known as the FOIL method for binomials) to multiply each term in the first expression by each term in the second expression step-by-step. Here is the detailed derivation:
1. Multiply the first terms of both binomials:
[tex]\[
5r \cdot 3r = 15r^2
\][/tex]
2. Multiply the outer terms:
[tex]\[
5r \cdot (-4) = -20r
\][/tex]
3. Multiply the inner terms:
[tex]\[
2 \cdot 3r = 6r
\][/tex]
4. Multiply the last terms:
[tex]\[
2 \cdot (-4) = -8
\][/tex]
Now, sum up all these results:
[tex]\[
15r^2 + (-20r) + 6r + (-8)
\][/tex]
Combine the like terms [tex]\(-20r\)[/tex] and [tex]\(6r\)[/tex]:
[tex]\[
15r^2 - 14r - 8
\][/tex]
Thus, the product of [tex]\((5r + 2)(3r - 4)\)[/tex] is:
[tex]\[
15r^2 - 14r - 8
\][/tex]
Therefore, the correct answer is:
[tex]\[
15r^2 - 14r - 8
\][/tex]
