The problem asks us to simplify the expression $\frac{(x^2+5x+4)(x-1)}{(x^2-1)}$.

1. Problem Description

The problem asks us to simplify the expression $\frac{(x^2+5x+4)(x-1)}{(x^2-1)}$.

2. Solution Steps

First, we factor the quadratic expressions in the numerator and denominator.

$x^2 + 5x + 4 = (x+1)(x+4)$

$x^2 - 1 = (x-1)(x+1)$

Substituting these factorizations into the expression yields:

$\frac{(x^2+5x+4)(x-1)}{(x^2-1)} = \frac{(x+1)(x+4)(x-1)}{(x-1)(x+1)}$

Now we can cancel out the common factors $(x+1)$ and $(x-1)$ from the numerator and denominator, provided $x \ne 1$ and $x \ne -1$.

$\frac{(x+1)(x+4)(x-1)}{(x-1)(x+1)} = x+4$

3. Final Answer

$x+4$