To solve the expression [tex]\((3x - 1)(12x + 4)\)[/tex] and expand it, you will follow these steps:
1. Distribute each term in the first binomial to every term in the second binomial. This process is often referred to as using the distributive property or applying the FOIL (First, Outer, Inner, Last) method for binomials:
[tex]\[
(3x - 1)(12x + 4) = (3x \cdot 12x) + (3x \cdot 4) + (-1 \cdot 12x) + (-1 \cdot 4)
\][/tex]
2. Calculate each product individually:
- First: Multiply the first terms of each binomial:
[tex]\[
3x \cdot 12x = 36x^2
\][/tex]
- Outer: Multiply the outer terms of the binomials:
[tex]\[
3x \cdot 4 = 12x
\][/tex]
- Inner: Multiply the inner terms of the binomials:
[tex]\[
-1 \cdot 12x = -12x
\][/tex]
- Last: Multiply the last terms of each binomial:
[tex]\[
-1 \cdot 4 = -4
\][/tex]
3. Sum all these products to form the expanded expression:
[tex]\[
36x^2 + 12x - 12x - 4
\][/tex]
4. Combine like terms in the expression:
- The [tex]\( +12x \)[/tex] and [tex]\( -12x \)[/tex] terms cancel each other out since they sum to zero.
[tex]\[
36x^2 + 12x - 12x - 4 = 36x^2 - 4
\][/tex]
Therefore, the expanded form of the expression [tex]\((3x - 1)(12x + 4)\)[/tex] is:
[tex]\[
36x^2 - 4
\][/tex]
