To find the product of the expressions [tex]\( (6r - 1) \)[/tex] and [tex]\( (-8r - 3) \)[/tex], we will use the distributive property of multiplication over addition, also known as the FOIL method for binomials. The steps are as follows:
1. First: Multiply the first terms in each binomial:
[tex]\[
6r \times (-8r) = -48r^2
\][/tex]
2. Outer: Multiply the outer terms in the product:
[tex]\[
6r \times (-3) = -18r
\][/tex]
3. Inner: Multiply the inner terms in the product:
[tex]\[
-1 \times (-8r) = 8r
\][/tex]
4. Last: Multiply the last terms in each binomial:
[tex]\[
-1 \times (-3) = 3
\][/tex]
Next, add these results together:
[tex]\[
-48r^2 + (-18r) + 8r + 3
\][/tex]
Combine like terms (the [tex]\( r \)[/tex] terms):
[tex]\[
-48r^2 + (-18r + 8r) + 3 = -48r^2 - 10r + 3
\][/tex]
Therefore, the product of [tex]\( (6r - 1)(-8r - 3) \)[/tex] is:
[tex]\[
-48r^2 - 10r + 3
\][/tex]
From the given options, the correct answer is:
[tex]\[
-48r^2 - 10r + 3
\][/tex]
