We are asked to simplify the expression $\frac{a+5}{a-1} \div (a^2+4a-5)$.

1. Problem Description

We are asked to simplify the expression $\frac{a+5}{a-1} \div (a^2+4a-5)$.

2. Solution Steps

First, we can rewrite the division as multiplication by the reciprocal:

$\frac{a+5}{a-1} \div (a^2+4a-5) = \frac{a+5}{a-1} \cdot \frac{1}{a^2+4a-5}$

Next, we can factor the quadratic expression $a^2+4a-5$:

$a^2+4a-5 = (a+5)(a-1)$

Now we can substitute this factorization back into the expression:

$\frac{a+5}{a-1} \cdot \frac{1}{(a+5)(a-1)}$

Then, we simplify the expression by canceling the common factor $(a+5)$:

$\frac{a+5}{a-1} \cdot \frac{1}{(a+5)(a-1)} = \frac{1}{a-1} \cdot \frac{1}{a-1}$

Finally, we multiply the two fractions:

$\frac{1}{a-1} \cdot \frac{1}{a-1} = \frac{1}{(a-1)^2}$

3. Final Answer

$\frac{1}{(a-1)^2}$