To solve this problem, let's begin by factoring each denominator in the given fractions.
Given fractions are:
[tex]\[
\frac{11x}{x^2 + 4x - 12} - \frac{7}{2x^2 - 4x}
\][/tex]
We need to factor the denominators:
1. For the first fraction:
[tex]\[
x^2 + 4x - 12
\][/tex]
The factors of this quadratic expression are:
[tex]\[
(x - 2)(x + 6)
\][/tex]
2. For the second fraction:
[tex]\[
2x^2 - 4x
\][/tex]
We can factor out the common factor of 2x:
[tex]\[
2x(x - 2)
\][/tex]
Now, to find the least common denominator (LCD), we need to collect all unique factors from the factored denominators.
From the first fraction:
[tex]\[
(x - 2), (x + 6)
\][/tex]
From the second fraction:
[tex]\[
2, x, (x - 2)
\][/tex]
To form the LCD, we gather each unique factor:
[tex]\[
2, x, (x - 2), (x + 6)
\][/tex]
Thus, the factors that compose the Least Common Denominator (LCD) are:
[tex]\[
2, x, (x - 2), (x + 6)
\][/tex]
Therefore, the correct answers are:
[tex]\[
2x, (x - 2), (x + 6)
\][/tex]
