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It appears that some parts of the text were misinterpreted during extraction, likely due to the quality of the image. Let me manually interpret and factor the correct equations from the provided image.

### Correctly Interpreted Equations for Factoring:

1. \( y^3 + 9y^2 \)
2. \( 5x^2 y^3 + 15x^3 y^2 \)
3. \( 12t^5 - 20t^4 + 8t^2 - 16 \)
4. \( p^2 - 36 \)
5. \( 25 - x^2 \)
6. \( 4a^3 - 49a \)
7. \( (a + b)^2 - 100 \)
8. \( 9 - (x - y)^2 \)
9. \( y^3 + 8 \)
10. \( 64y^4 + y \)

Let’s go ahead and factor them step by step.

### Solutions:
1. **Factor** \( y^3 + 9y^2 \):
   \[
   y^3 + 9y^2 = y^2(y + 9)
   \]

2. **Factor** \( 5x^2 y^3 + 15x^3 y^2 \):
   \[
   5x^2 y^3 + 15x^3 y^2 = 5x^2 y^2 (y + 3x)
   \]

3. **Factor** \( 12t^5 - 20t^4 + 8t^2 - 16 \):
   Group terms and factor by grouping:
   \[
   12t^5 - 20t^4 + 8t^2 - 16 = 4t^2(3t^3 - 5t^2 + 2) - 16
   \]
   At this point, no further factoring is possible.

4. **Factor** \( p^2 - 36 \):
   This is a difference of squares:
   \[
   p^2 - 36 = (p - 6)(p + 6)
   \]

5. **Factor** \( 25 - x^2 \):
   This is also a difference of squares:
   \[
   25 - x^2 = (5 - x)(5 + x)
   \]

6. **Factor** \( 4a^3 - 49a \):
   Factor out \( a \):
   \[
   4a^3 - 49a = a(4a^2 - 49) = a(2a - 7)(2a + 7)
   \]
   (Difference of squares)

7. **Factor** \( (a + b)^2 - 100 \):
   This is a difference of squares:
   \[
   (a + b)^2 - 100 = [(a + b) - 10][(a + b) + 10]
   \]

8. **Factor** \( 9 - (x - y)^2 \):
   This is a difference of squares:
   \[
   9 - (x - y)^2 = (3 - (x - y))(3 + (x - y)) = (3 - x + y)(3 + x - y)
   \]

9. **Factor** \( y^3 + 8 \):
   This is a sum of cubes:
   \[
   y^3 + 8 = (y + 2)(y^2 - 2y + 4)
   \]

10. **Factor** \( 64y^4 + y \):
   Factor out \( y \):
   \[
   64y^4 + y = y(64y^3 + 1)
   \]

Would you like further details or explanations on any specific factorization? Here are some related questions to explore further:

---

1. How do you factor the difference of squares?
2. What are the steps for factoring a sum of cubes like \( y^3 + 8 \)?
3. How do you factor by grouping, as in \( 12t^5 - 20t^4 + 8t^2 - 16 \)?
4. What is the factorization of higher powers, like \( y^3 + 9y^2 \)?
5. How can you recognize when a trinomial is a perfect square?

**Tip:** Always check for common factors before applying advanced techniques like difference of squares or sum/difference of cubes.

Factor the following algebraic expressions.

Learn how to factor various algebraic expressions such as polynomials, difference of squares, and sum of cubes. This guide covers core algebraic concepts and provides step-by-step solutions for each equation. Suitable for students in Grades 8-10.

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