To find the product of the binomials [tex]\((4a - 1)\)[/tex] and [tex]\((2b + 3)\)[/tex], we use the distributive property, often referred to as the FOIL method for binomials. FOIL stands for First, Outer, Inner, and Last, representing the way we multiply each term in the first binomial by each term in the second binomial.
Let's do this step-by-step:
1. First: Multiply the first terms in each binomial.
[tex]$ 4a \cdot 2b = 8ab $[/tex]
2. Outer: Multiply the outer terms in the binomials.
[tex]$ 4a \cdot 3 = 12a $[/tex]
3. Inner: Multiply the inner terms in the binomials.
[tex]$ -1 \cdot 2b = -2b $[/tex]
4. Last: Multiply the last terms in each binomial.
[tex]$ -1 \cdot 3 = -3 $[/tex]
Now, we combine all these products:
[tex]$ 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 $[/tex]
Among the given choices, the correct one is:
[tex]$ \boxed{8ab + 12a - 2b - 3} $[/tex]
