To determine the quotient of the given expression:
[tex]\[
\frac{x^2-4}{(x-2)(x-4)}
\][/tex]
we need to simplify it step-by-step.
1. Factor the numerator: The numerator [tex]\( x^2 - 4 \)[/tex] is a difference of squares, which we can factor as follows:
[tex]\[
x^2 - 4 = (x + 2)(x - 2)
\][/tex]
2. Rewrite the expression with the factored numerator:
[tex]\[
\frac{(x + 2)(x - 2)}{(x - 2)(x - 4)}
\][/tex]
3. Cancel common factors: We see that [tex]\( (x - 2) \)[/tex] is present in both the numerator and the denominator. So, we can cancel out [tex]\( (x - 2) \)[/tex]:
[tex]\[
\frac{(x + 2) \cancel{(x - 2)}}{\cancel{(x - 2)}(x - 4)} = \frac{x + 2}{x - 4}
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\frac{x + 2}{x - 4}
\][/tex]
Among the provided options, this matches option A:
[tex]\[
\boxed{\frac{x+2}{x-4}}
\][/tex]
Thus, the correct answer is:
A. [tex]\(\frac{x+2}{x-4}\)[/tex]
