(x ^ 2 + 2x - 3)/(x ^ 2 - 2x - 3) * (3 - x)/(3 + x)

We are tasked with simplifying the following expression:

\[
\frac{x^2 + 2x - 3}{x^2 - 2x - 3} \times \frac{3 - x}{3 + x}
\]

### Step 1: Factor each quadratic expression

1. **Factor the numerator of the first fraction** \((x^2 + 2x - 3)\):
   \[
   x^2 + 2x - 3 = (x + 3)(x - 1)
   \]

2. **Factor the denominator of the first fraction** \((x^2 - 2x - 3)\):
   \[
   x^2 - 2x - 3 = (x - 3)(x + 1)
   \]

So, the expression becomes:

\[
\frac{(x + 3)(x - 1)}{(x - 3)(x + 1)} \times \frac{3 - x}{3 + x}
\]

### Step 2: Simplify the second fraction

The term \((3 - x)\) can be rewritten as \(-(x - 3)\), so:

\[
\frac{3 - x}{3 + x} = \frac{-(x - 3)}{x + 3}
\]

Now the expression becomes:

\[
\frac{(x + 3)(x - 1)}{(x - 3)(x + 1)} \times \frac{-(x - 3)}{x + 3}
\]

### Step 3: Cancel common factors

- The term \((x + 3)\) in the numerator of the first fraction cancels with \((x + 3)\) in the denominator of the second fraction.
- The term \((x - 3)\) in the denominator of the first fraction cancels with \((x - 3)\) in the numerator of the second fraction.

Thus, the expression simplifies to:

\[
\frac{-(x - 1)}{x + 1}
\]

### Final Answer:
\[
\frac{-(x - 1)}{x + 1} \quad \text{or} \quad \frac{1 - x}{x + 1}
\]

Would you like more details or have any questions?

Here are five related questions to explore this topic further:
1. How do you factor quadratic expressions like \(x^2 + 2x - 3\)?
2. What is the general rule for simplifying rational expressions?
3. How do you recognize and cancel common factors in rational expressions?
4. What is the importance of factoring in simplifying algebraic expressions?
5. Can this expression have any restrictions on \(x\) based on the denominator?

**Tip:** Always check for factors that can be canceled before multiplying rational expressions—it simplifies the process!

Learn how to simplify the expression (x^2 + 2x - 3)/(x^2 - 2x - 3) * (3 - x)/(3 + x) through factoring quadratics and rational expressions. This step-by-step guide explains how to factor, simplify, and cancel common factors. Suitable for students in Grades 9-11.

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