https://file.aiquickdraw.com/mathbot/file/17282075019166glx3cm0.jpg

To solve the matching problem, let's analyze both columns step by step.

### Column-I:
(A) \( 3a^5 - 48a^3 \)  
(B) \( 2a^5 - 32a \)  
(C) \( 4(xy + 1)^2 - 9(x - 1)^2 \)  
(D) \( xy^9 - yx^9 \)

### Column-II:
(p) \( 2a(a - 2)(a + 2)(a^2 + 4) \)  
(q) \( (2xy + 3x - 1)(2xy - 3x + 5) \)  
(r) \( -xy(x^2 + y^2)(x + y)(x - y)(x^4 + y^4) \)  
(s) \( 3a^3(a - 4)(a + 4) \)

### Matching:

- **(A) \( 3a^5 - 48a^3 \)**: Factor out the common terms:  
  \( 3a^3(a^2 - 16) = 3a^3(a - 4)(a + 4) \)  
  So, match this with **(s)**.

- **(B) \( 2a^5 - 32a \)**: Factor out the common terms:  
  \( 2a(a^4 - 16) = 2a(a - 2)(a + 2)(a^2 + 4) \)  
  So, match this with **(p)**.

- **(C) \( 4(xy + 1)^2 - 9(x - 1)^2 \)**: This is in the form of a difference of squares:  
  \( [2(xy + 1)]^2 - [3(x - 1)]^2 = (2xy + 3x - 1)(2xy - 3x + 5) \)  
  So, match this with **(q)**.

- **(D) \( xy^9 - yx^9 \)**: Factor out \( xy \):  
  \( xy(y^8 - x^8) = xy(x^4 + y^4)(x^2 + y^2)(x + y)(x - y) \)  
  So, match this with **(r)**.

### Final Answer:
- (A) matches with (s)
- (B) matches with (p)
- (C) matches with (q)
- (D) matches with (r)

Let me know if you'd like further details on any of these! 

### Relative Questions:
1. Can you explain the factorization technique used in matching \( 3a^5 - 48a^3 \)?
2. How can you generalize the difference of squares formula for expressions like \( 4(xy + 1)^2 - 9(x - 1)^2 \)?
3. Why is \( xy^9 - yx^9 \) simplified as a factorization problem?
4. How do higher powers (like \( a^5 \) or \( y^9 \)) influence the factorization?
5. What other techniques can be applied when dealing with complex polynomial factorizations?

### Tip:
Always look for common factors or recognizable patterns like the difference of squares or sum/difference of cubes when factoring.

Match the Columns: Factor and match terms in Column-I with those in Column-II.

Solve a polynomial factorization problem by matching expressions in Column-I with their factored forms in Column-II. Learn about common factors, the difference of squares, and advanced polynomial factorization techniques. Suitable for students in Grades 9-11.

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