Sure, let's go through the solution step-by-step to determine which expression is equivalent to the given expression [tex]\(\frac{\frac{1}{a^2 - 2a - 8}}{\frac{a - 6}{a - 4}}\)[/tex].
Step 1: Rewrite the expression by multiplying by the reciprocal of the denominator.
[tex]\[
\frac{\frac{1}{a^2 - 2a - 8}}{\frac{a - 6}{a - 4}} = \frac{1}{a^2 - 2a - 8} \times \frac{a - 4}{a - 6}
\][/tex]
Step 2: Factor the quadratic expression in the denominator of the first fraction.
[tex]\[
a^2 - 2a - 8 = (a - 4)(a + 2)
\][/tex]
Step 3: Substitute the factored form into the expression.
[tex]\[
\frac{1}{(a - 4)(a + 2)} \times \frac{a - 4}{a - 6}
\][/tex]
Step 4: Cancel out the common factor [tex]\((a - 4)\)[/tex] from the numerator and the denominator.
[tex]\[
\frac{1}{(a + 2)} \times \frac{1}{a - 6} = \frac{1}{(a + 2)(a - 6)}
\][/tex]
Now we have the simplified equivalent expression:
[tex]\[
\frac{1}{(a + 2)(a - 6)}
\][/tex]
Based on the choices provided, the correct expression that matches our simplified result is:
[tex]\[
\boxed{\frac{1}{(a + 2)(a - 6)}}
\][/tex]
Hence, the correct answer is the first choice:
[tex]\[
\frac{1}{(a + 2)(a - 6)}
\][/tex]
