To solve the problem of finding the least common denominator (LCD) for the given fractions, let's go through the solution step by step.
First, we need to factorize the denominators:
1. The first denominator is [tex]\( x^2 + 4x - 12 \)[/tex].
To factorize [tex]\( x^2 + 4x - 12 \)[/tex], we look for two numbers that multiply to [tex]\(-12\)[/tex] and add to [tex]\(4\)[/tex]. These numbers are [tex]\(6\)[/tex] and [tex]\(-2\)[/tex]. So, we can factorize it as:
[tex]\[
x^2 + 4x - 12 = (x - 2)(x + 6)
\][/tex]
2. The second denominator is [tex]\( 2x^2 - 4x \)[/tex].
To factorize [tex]\( 2x^2 - 4x \)[/tex], we can factor out the common factor [tex]\(2x\)[/tex]:
[tex]\[
2x^2 - 4x = 2x(x - 2)
\][/tex]
Next, to find the LCD, we take the product of all unique factors from both denominators.
3. From our factorization, the factors of [tex]\( x^2 + 4x - 12 \)[/tex] are:
[tex]\[
(x - 2)(x + 6)
\][/tex]
4. The factors of [tex]\( 2x^2 - 4x \)[/tex] are:
[tex]\[
2x(x - 2)
\][/tex]
The unique factors from both factorizations are:
[tex]\[
(x - 2), (x + 6), 2x
\][/tex]
Therefore, the least common denominator (LCD) is composed of these unique factors.
Hence, the correct answers are:
- [tex]\((x-2)\)[/tex]
- [tex]\((x+6)\)[/tex]
- [tex]\(2x\)[/tex]
The factors that compose the least common denominator are:
[tex]\[
\boxed{(x-2), (x+6), 2x}
\][/tex]
