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Let's analyze the given polynomials:

We are given four polynomials \( p(x), q(x), r(x), \) and \( s(x) \):

\[
p(x) = x^2 - 5x - 6, \quad q(x) = x + 1, \quad r(x) = 2x^3 - 4x^2 - 6x
\]
and 
\[
s(x) = p(x)q(x)r(x)
\]

### Statement 1: The degree of \( p(x) + q(x) \) is 3.
- **Solution**: The degree of \( p(x) \) is 2 (since \( p(x) = x^2 - 5x - 6 \)) and the degree of \( q(x) \) is 1 (since \( q(x) = x + 1 \)).
- When adding two polynomials, the degree of the sum is the highest degree of the individual polynomials, provided they do not cancel out. Here, \( \deg(p(x)) = 2 \) and \( \deg(q(x)) = 1 \). Thus, the degree of \( p(x) + q(x) \) is **2**.
- This statement is **false**.

### Statement 2: The degree of \( p(x)r(x) \) is 5.
- **Solution**: The degree of \( p(x) \) is 2, and the degree of \( r(x) \) is 3. When multiplying polynomials, the degrees add. So, the degree of \( p(x)r(x) \) is \( 2 + 3 = 5 \).
- This statement is **true**.

### Statement 3: When \( p(x) \) divides \( r(x) \), the obtained remainder is a linear function.
- **Solution**: We perform polynomial division of \( r(x) \) by \( p(x) \). First, \( r(x) = 2x^3 - 4x^2 - 6x \) and \( p(x) = x^2 - 5x - 6 \). Performing the division step-by-step, we find that the remainder is a **constant function** (not linear).
- This statement is **false**.

### Statement 4: When \( p(x) \) divides \( r(x) \), the obtained remainder is a quadratic function.
- **Solution**: As we saw in Statement 3, the remainder is a constant, not a quadratic function.
- This statement is **false**.

### Final Answer:
- The only correct statement is **Statement 2**: "The degree of \( p(x)r(x) \) is 5."

Would you like further details on any of the steps? Here are 5 related questions:

1. How do you perform polynomial division between \( r(x) \) and \( p(x) \)?
2. What are the general rules for adding and multiplying polynomial degrees?
3. Can we always predict the degree of the remainder in polynomial division?
4. How does polynomial division affect the behavior of the divisor and dividend?
5. What happens when we multiply or add polynomials with similar degrees?

**Tip:** When dividing polynomials, focus on the leading terms to simplify the division process step by step.

Consider four polynomials p(x), q(x), r(x), and s(x) as follows: p(x) = x^2 - 5x - 6, q(x) = x + 1, r(x) = 2x^3 - 4x^2 - 6x, s(x) = p(x)q(x)r(x). Which of the following options is/are true? 1) The degree of p(x) + q(x) is 3. 2) The degree of p(x)r(x) is 5. 3) When p(x) divides r(x), the remainder is a linear function. 4) When p(x) divides r(x), the remainder is a quadratic function.

Explore polynomial operations including addition, multiplication, and division. Understand how to determine the degree of polynomial expressions like p(x) + q(x) and p(x)r(x), and find out the nature of remainders in polynomial division. Suitable for Grades 10-12.

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