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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.

Simplify the expression 6/(x-1) - 2/(x+1) divided by (x/(x-1) + 1/(x+1)), and give your answer in the form of f(x)/g(x).

Learn how to simplify a complex rational expression involving fractions with variable denominators. This problem is ideal for students in grades 9-12, focusing on algebraic concepts, including rational expressions and factoring.

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