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To simplify the given expression:

\[
\frac{x + 1}{x - 1} \cdot \frac{x^2 - 1}{x^2 + 2x + 1}
\]

Let's break this down step-by-step.

### Step 1: Factorize the expressions
1. **Numerator of the second fraction**: \( x^2 - 1 \) is a difference of squares, which can be factored as:
   \[
   x^2 - 1 = (x - 1)(x + 1)
   \]

2. **Denominator of the second fraction**: \( x^2 + 2x + 1 \) is a perfect square trinomial, which can be factored as:
   \[
   x^2 + 2x + 1 = (x + 1)^2
   \]

Now, the expression becomes:
\[
\frac{x + 1}{x - 1} \cdot \frac{(x - 1)(x + 1)}{(x + 1)^2}
\]

### Step 2: Simplify the expression
1. Combine the fractions:
   \[
   \frac{(x + 1)(x - 1)(x + 1)}{(x - 1)(x + 1)^2}
   \]

2. Cancel out the common terms:
   - \( (x + 1) \) in the numerator and \( (x + 1)^2 \) in the denominator reduce to:
   \[
   \frac{(x - 1)}{(x + 1)}
   \]

   - \( (x - 1) \) in the numerator and denominator also cancel each other out:
   \[
   1
   \]

So, the simplified form of the expression is:
\[
1
\]

### Final Answer
The expression simplifies to 1.

Would you like more details or have any questions about this process?

### Related Questions
1. How do you factor a difference of squares?
2. How do you simplify rational expressions with common terms?
3. What are the steps to factor a perfect square trinomial?
4. How do you multiply two rational expressions?
5. Why is it important to cancel terms in rational expressions?

### Tip
Always check for common factors in numerators and denominators before simplifying expressions. This often helps in reducing the expression quickly.

(x + 1)/(x - 1) * (x^2 - 1)/(x^2 + 2x + 1)

Learn how to simplify the rational expression (x + 1)/(x - 1) * (x^2 - 1)/(x^2 + 2x + 1) using factoring techniques. This solution covers key algebraic concepts such as difference of squares and perfect square trinomials, suitable for students in Grades 9-12.

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