To find the product of the binomials [tex]\((4a - 1)\)[/tex] and [tex]\((2b + 3)\)[/tex], we need to use the distributive property (also known as the distributive law of multiplication). This law states that each term in the first binomial must be multiplied by each term in the second binomial.
Let’s break it down step-by-step:
1. Multiply [tex]\(4a\)[/tex] by [tex]\(2b\)[/tex]:
[tex]\[
4a \cdot 2b = 8ab
\][/tex]
2. Multiply [tex]\(4a\)[/tex] by [tex]\(3\)[/tex]:
[tex]\[
4a \cdot 3 = 12a
\][/tex]
3. Multiply [tex]\(-1\)[/tex] by [tex]\(2b\)[/tex]:
[tex]\[
-1 \cdot 2b = -2b
\][/tex]
4. Multiply [tex]\(-1\)[/tex] by [tex]\(3\)[/tex]:
[tex]\[
-1 \cdot 3 = -3
\][/tex]
Now, combine all these results together:
[tex]\[
(4a - 1) \cdot (2b + 3) = 8ab + 12a - 2b - 3
\][/tex]
Therefore, the product of the binomials [tex]\((4a - 1)\)[/tex] and [tex]\((2b + 3)\)[/tex] is:
[tex]\[
8ab + 12a - 2b - 3
\][/tex]
The correct answer from the given options is:
[tex]\[
8 a b + 12 a - 2 b - 3
\][/tex]
