To subtract the fractions [tex]\(\frac{4}{x-2}\)[/tex] and [tex]\(\frac{4}{x+6}\)[/tex], follow these steps:
1. Find a common denominator: The fractions have different denominators, so you need to find a common denominator. For the fractions [tex]\(\frac{4}{x-2}\)[/tex] and [tex]\(\frac{4}{x+6}\)[/tex], the common denominator will be [tex]\((x-2)(x+6)\)[/tex].
2. Create equivalent fractions with the common denominator:
- For [tex]\(\frac{4}{x-2}\)[/tex], multiply both the numerator and the denominator by [tex]\((x+6)\)[/tex]:
[tex]\[
\frac{4}{x-2} = \frac{4(x+6)}{(x-2)(x+6)}
\][/tex]
- For [tex]\(\frac{4}{x+6}\)[/tex], multiply both the numerator and the denominator by [tex]\((x-2)\)[/tex]:
[tex]\[
\frac{4}{x+6} = \frac{4(x-2)}{(x-2)(x+6)}
\][/tex]
3. Subtract the fractions:
- Now, with the common denominator, subtract the numerators:
[tex]\[
\frac{4(x+6)}{(x-2)(x+6)} - \frac{4(x-2)}{(x-2)(x+6)}
\][/tex]
Combine the fractions:
[tex]\[
\frac{4(x+6) - 4(x-2)}{(x-2)(x+6)}
\][/tex]
4. Simplify the numerator:
- Distribute the 4 in the numerators:
[tex]\[
\frac{4x + 24 - 4x + 8}{(x-2)(x+6)}
\][/tex]
- Combine like terms:
[tex]\[
\frac{32}{(x-2)(x+6)}
\][/tex]
So, after performing the subtraction and simplification, the resulting fraction is:
[tex]\[
\frac{32}{(x-2)(x+6)}
\][/tex]
Thus, the correct answer is:
C. [tex]\(\frac{32}{(x-2)(x+6)}\)[/tex]
