The problem asks to simplify the expression:

$\frac{\frac{6}{x - 1} - \frac{2}{x + 1}}{\frac{x}{x - 1} + \frac{1}{x + 1}}$

and provide the result in the form of $\frac{f(x)}{g(x)}$, where $f(x)$ and $g(x)$ are simplified functions.

Let's break down the solution step-by-step.

Step 1: Simplify the Numerator

The numerator is: $\frac{6}{x - 1} - \frac{2}{x + 1}$ To combine these, we need a common denominator, which is $(x - 1)(x + 1)$.

$\frac{6(x + 1) - 2(x - 1)}{(x - 1)(x + 1)}$

Expanding the terms in the numerator: $\frac{6x + 6 - 2x + 2}{(x - 1)(x + 1)} = \frac{4x + 8}{(x - 1)(x + 1)}$

Simplify the numerator by factoring out a 4: $\frac{4(x + 2)}{(x - 1)(x + 1)}$

Step 2: Simplify the Denominator

The denominator is: $\frac{x}{x - 1} + \frac{1}{x + 1}$ Using a common denominator of $(x - 1)(x + 1)$, we get:

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

Expanding the terms in the numerator: $\frac{x^2 + x + x - 1}{(x - 1)(x + 1)} = \frac{x^2 + 2x - 1}{(x - 1)(x + 1)}$

Step 3: Divide the Numerator by the Denominator

Now we have:

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

Final Answer

So, in the form $\frac{f(x)}{g(x)}$, we have: $f(x) = 4(x + 2)$ $g(x) = x^2 + 2x - 1$

Would you like further explanation on any part of this solution?

Here are some related questions:

1. How do you find a common denominator in rational expressions?
2. What are the steps to simplify complex fractions?
3. Why do we factor out common terms in expressions?
4. How can you identify the form of the answer in fraction-based questions?
5. What are the rules for simplifying expressions with variables?

Tip: When simplifying expressions, always look for common factors to cancel out after combining terms.