To multiply the expressions [tex]\((3r - 1)\)[/tex] and [tex]\((4r + 3)\)[/tex], we can use the distributive property (also known as the FOIL method for binomials). Here’s the step-by-step process:
1. Apply the distributive property for each term in the first binomial to each term in the second binomial (FOIL stands for First, Outer, Inner, Last):
- First: Multiply the first terms in each binomial:
[tex]\[
(3r) \cdot (4r) = 12r^2
\][/tex]
- Outer: Multiply the outer terms in the multiplication:
[tex]\[
(3r) \cdot (3) = 9r
\][/tex]
- Inner: Multiply the inner terms:
[tex]\[
(-1) \cdot (4r) = -4r
\][/tex]
- Last: Multiply the last terms in each binomial:
[tex]\[
(-1) \cdot (3) = -3
\][/tex]
2. Combine all these results together:
[tex]\[
12r^2 + 9r - 4r - 3
\][/tex]
3. Simplify by combining like terms (in this case, the [tex]\(r\)[/tex] terms):
[tex]\[
12r^2 + (9r - 4r) - 3 = 12r^2 + 5r - 3
\][/tex]
Therefore,
[tex]\[
(3r - 1)(4r + 3) = 12r^2 + 5r - 3
\][/tex]
This is the simplified form of the expression.
