We want to expand \(\left( \frac{1 - x}{1 + x} \right)^n\) up to the \(x^2\) term. The best approach is to express this as a binomial series.

### Step 1: Rewrite the expression

We begin by rewriting the expression as follows:

\[
\left( \frac{1 - x}{1 + x} \right)^n = \left( (1 - x)(1 + x)^{-1} \right)^n = (1 - x)^n (1 + x)^{-n}
\]

Now, we can expand both \((1 - x)^n\) and \((1 + x)^{-n}\) separately using the binomial theorem.

### Step 2: Expand \((1 - x)^n\)

The binomial expansion of \((1 - x)^n\) is:

\[
(1 - x)^n = 1 - nx + \frac{n(n-1)}{2}x^2 + \cdots
\]

### Step 3: Expand \((1 + x)^{-n}\)

The binomial expansion of \((1 + x)^{-n}\) is:

\[
(1 + x)^{-n} = 1 - nx + \frac{n(n+1)}{2}x^2 + \cdots
\]

### Step 4: Multiply the two expansions

Now, we multiply the two expansions up to the \(x^2\) term:

\[
\left( 1 - nx + \frac{n(n-1)}{2}x^2 \right) \times \left( 1 - nx + \frac{n(n+1)}{2}x^2 \right)
\]

We only need to consider terms up to \(x^2\):

- Constant term: \(1 \times 1 = 1\)
- Linear term: \((-nx) \times 1 + 1 \times (-nx) = -2nx\)
- Quadratic term:
  \[
  (-nx) \times (-nx) + \left(\frac{n(n-1)}{2}x^2\right) \times 1 + 1 \times \left(\frac{n(n+1)}{2}x^2\right)
  \]
  \[
  = n^2x^2 + \frac{n(n-1)}{2}x^2 + \frac{n(n+1)}{2}x^2
  \]
  \[
  = n^2x^2 + \frac{n(n-1) + n(n+1)}{2}x^2 = n^2x^2 + \frac{2n^2}{2}x^2 = n^2x^2
  \]

### Step 5: Final result

Thus, the expansion of \(\left( \frac{1 - x}{1 + x} \right)^n\) up to the \(x^2\) term is:

\[
\left( \frac{1 - x}{1 + x} \right)^n = 1 - 2nx + n^2x^2 + \cdots
\]

Would you like further details on any part of this process?

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Here are five questions related to this topic:
1. How is the binomial theorem applied to expand terms like \((1-x)^n\)?
2. What is the significance of limiting the expansion to the \(x^2\) term?
3. How do higher-order terms affect the overall accuracy of the expansion?
4. Can this expansion be used for complex numbers or is it limited to real numbers?
5. How does the series behave as \(n\) increases?

**Tip:** When multiplying expansions, keep track of each term's degree to ensure you're only expanding to the required order.

Learn how to expand ((1-x)/(1+x))^n up to the x^2 term using binomial expansion. This solution covers key algebraic concepts and the application of the binomial theorem. Ideal for students in Grades 10-12.

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