Let's solve the expression step by step:

Given expression:

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

Step 1: Factorize the denominator in the first fraction

The first fraction has a denominator $x^2 - 1$, which is a difference of squares. We can factorize it:

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

So, the expression becomes:

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

Step 2: Combine the fractions

To subtract these fractions, we need a common denominator. The first fraction already has $(x - 1)(x + 1)$ as the denominator. To make the second fraction have the same denominator, multiply the numerator and denominator of the second fraction by $(x + 1)$:

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

Simplifying the numerator of the second fraction:

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

So the expression is now:

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

Step 3: Simplify the numerator

Distribute the subtraction across the terms in the numerator:

$4x - x^2 - 2x - 1 = -x^2 + 2x - 1$

So, the expression simplifies to:

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

Final Answer:

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

This is the simplified form of the given expression.

Would you like more details, or do you have any questions?

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Here are five related questions to expand your understanding:

1. How do you factor other types of polynomials, such as $x^2 + 5x + 6$?
2. What are the general steps to simplify a complex rational expression?
3. How can you identify and factor differences of squares in algebraic expressions?
4. Can you explain how to multiply and divide rational expressions?
5. What is the importance of finding a common denominator when adding or subtracting fractions?

Tip: Always factorize where possible in algebraic expressions; it simplifies the problem and helps to avoid mistakes.