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The image asks to find the common factor and factorize the given polynomials to their maximum expression. Let's go through each polynomial provided in the image:

1. **\( 9x^4 + 99x^3 \)**:
   - The common factor between the two terms is \( 9x^3 \).
   - Factorization: \( 9x^3(x + 11) \).

2. **\( -288x^2y + 162xy^2 + 128x^3 \)**:
   - The common factor is \( 2x \).
   - Factorization: \( 2x(-144xy + 81y^2 + 64x^2) \).

3. **\( 81x^2y - 25y \)**:
   - The common factor is \( y \).
   - Factorization: \( y(81x^2 - 25) \), which is a difference of squares.
   - Further factorization: \( y(9x + 5)(9x - 5) \).

4. **\( 10s^3t^3 + st^6 + 25s^5 \)**:
   - The common factor is \( s \).
   - Factorization: \( s(10s^2t^3 + t^6 + 25s^4) \).

5. **\( 90rt^2 + 40rs^2 - 120rst \)**:
   - The common factor is \( 10r \).
   - Factorization: \( 10r(9t^2 + 4s^2 - 12st) \).

6. **\( -121a^5b^2 + \frac{49}{25}a^7 \)**:
   - The common factor is \( a^5 \).
   - Factorization: \( a^5\left( -121b^2 + \frac{49}{25}a^2 \right) \).

These are the factorizations of the polynomials provided. Let me know if you would like any further details or have any questions.

### Relative Questions:
1. How do you identify the greatest common factor (GCF) in a polynomial?
2. What is the difference of squares, and how do you factor it?
3. Why is it important to factor polynomials?
4. How can you verify if a factorization is correct?
5. What happens if a polynomial has no common factor?

### Tip:
When factoring polynomials, always start by identifying the greatest common factor (GCF) before attempting more complex factorization methods.

Find the common factor and factorize the given polynomials to their maximum expression.

Learn how to find the common factor and factorize polynomials, such as 9x^4 + 99x^3 and 81x^2y - 25y. This problem explores algebraic concepts like the greatest common factor and the difference of squares, suitable for high school students (Grades 10-12).

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