The expression you provided is:

$\frac{x^4 + x^3 + x^2 - x - 2}{x^2 - 1}$

This can be simplified by factoring the denominator and trying to factor or simplify the numerator. Let’s break it down:

Step 1: Factor the denominator

The denominator is $x^2 - 1$, which is a difference of squares: $x^2 - 1 = (x - 1)(x + 1)$

Step 2: Factor the numerator

Now, let's attempt to factor the numerator $x^4 + x^3 + x^2 - x - 2$. We'll look for potential factors that match the denominator.

To factor the numerator, we can try factoring by grouping or use synthetic division to check if either $x - 1$ or $x + 1$ is a factor.

After applying synthetic division or factoring methods, we find that the numerator can be factored as:

$x^4 + x^3 + x^2 - x - 2 = (x^2 + 1)(x^2 - 1)$

Step 3: Simplify the expression

Now substitute the factorized forms of both the numerator and denominator:

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

Notice that $x^2 - 1$ can be factored into $(x - 1)(x + 1)$, so we can cancel out the common factors of $(x - 1)(x + 1)$ from both the numerator and the denominator:

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

Final simplified form:

The expression simplifies to:

$x^2 + 1$

Conclusion:

The simplified form of the given expression is:

$x^2 + 1$

Would you like more details or further clarification on any step?

Here are 5 related questions to explore:

1. How do you factor the difference of squares in expressions like $x^2 - 1$?
2. What is synthetic division, and how can it help in factoring polynomials?
3. How do you determine whether a polynomial can be factored by grouping?
4. What are some strategies to simplify rational expressions like the one in the problem?
5. What happens to the expression if $x = 1$ or $x = -1$?

Tip: When simplifying rational expressions, always check for common factors in both the numerator and denominator to cancel them out efficiently.