Simplify the expression $\frac{(x^2+5x+4)(x-1)}{(x^2-1)}$.

1. Problem Description

Simplify the expression $\frac{(x^2+5x+4)(x-1)}{(x^2-1)}$.

2. Solution Steps

First, factor the quadratic expression $x^2 + 5x + 4$. We are looking for two numbers that multiply to 4 and add to

5. These numbers are 1 and

4. Therefore, $x^2 + 5x + 4 = (x+1)(x+4)$.

Next, factor the difference of squares $x^2-1$. This is $x^2 - 1^2 = (x-1)(x+1)$.

Now, substitute the factored expressions into the original expression:

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

We can cancel 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

$x+4$