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Let's break down and analyze the mathematical work in the image. 

### Part 1 (Multiplication of Rational Expressions):

The first part involves simplifying the expression:

\[
\frac{r(p^2 - q^2)}{p^2 - 2pq + q^2} \times \frac{p^2 - pq - p + q}{p^2 - rp + pq - rq}
\]

#### Step-by-Step Process:
1. **Factor the terms**: 
   - The first fraction:
     \[
     \frac{r(p^2 - q^2)}{(p - q)(p - q)} = \frac{r(p - q)(p + q)}{(p - q)(p - q)}
     \]
   - The second fraction:
     \[
     \frac{p^2 - pq - p + q}{p^2 - rp + pq - rq} = \frac{(p - 1)(p - q)}{(p - 1)(p + q)}
     \]

2. **Simplify the product**: 
   Combine the two fractions:
   \[
   \frac{r(p - q)(p + q)}{(p - q)(p - q)} \times \frac{(p - 1)(p - q)}{(p - 1)(p + q)}
   \]

3. **Cancel common terms**:
   - \((p - q)\) appears in both the numerator and the denominator, so it cancels out.
   - \((p + q)\) also cancels out.
   - This leaves:
   \[
   \frac{r}{r} = 1
   \]

Thus, the simplified expression is:

\[
1
\]

---

### Part 2 (Simplification of Rational Expression):

The second part asks you to simplify:

\[
\frac{p - 1}{p^2 + 3p + 2} + \frac{2}{p + 1}
\]

#### Step-by-Step Process:
1. **Factor the quadratic** in the denominator of the first term:
   \[
   p^2 + 3p + 2 = (p + 1)(p + 2)
   \]
   Now the expression becomes:
   \[
   \frac{p - 1}{(p + 1)(p + 2)} + \frac{2}{p + 1}
   \]

2. **Find a common denominator**: 
   The common denominator between the two fractions is \((p + 1)(p + 2)\), so rewrite the second fraction with this denominator:
   \[
   \frac{2}{p + 1} = \frac{2(p + 2)}{(p + 1)(p + 2)}
   \]

3. **Combine the fractions**:
   \[
   \frac{p - 1}{(p + 1)(p + 2)} + \frac{2(p + 2)}{(p + 1)(p + 2)}
   \]
   \[
   = \frac{(p - 1) + 2(p + 2)}{(p + 1)(p + 2)}
   \]

4. **Simplify the numerator**:
   Expand the terms in the numerator:
   \[
   (p - 1) + 2(p + 2) = p - 1 + 2p + 4 = 3p + 3
   \]
   So the expression becomes:
   \[
   \frac{3(p + 1)}{(p + 1)(p + 2)}
   \]

5. **Cancel common terms**:
   The \((p + 1)\) terms cancel out, leaving:
   \[
   \frac{3}{p + 2}
   \]

Thus, the simplified expression is:

\[
\frac{3}{p + 2}
\]

---

### Final Answers:
- **First part**: \( 1 \)
- **Second part**: \( \frac{3}{p + 2} \)

---

Do you have any questions or would you like further details?

### Related Questions:
1. Can you explain how to factor quadratic expressions like \( p^2 + 3p + 2 \)?
2. What are the common strategies for simplifying rational expressions?
3. How can we handle more complex polynomials in rational expressions?
4. Why do certain terms cancel out when multiplying rational expressions?
5. Can these methods be applied to non-quadratic polynomials?

### Tip:
When simplifying rational expressions, always factor both the numerator and denominator first to identify common terms that can be canceled.

Write all these correctly and find perfect answers

Learn how to simplify complex rational expressions involving factoring quadratics and common term cancellation. This example covers the difference of squares, factoring quadratic expressions, and simplifying rational expressions. Suitable for Grades 10-12.

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