https://file.aiquickdraw.com/mathbot/file/1729794252814ce4b67ue.jpg

Let's work through factoring each of these polynomial expressions:

### 1. \( y^2 + 8y + 16 \)
This is a perfect square trinomial. We need two numbers that multiply to 16 and add to 8. These numbers are 4 and 4, so:
\[
y^2 + 8y + 16 = (y + 4)(y + 4) = (y + 4)^2
\]

### 2. \( y^2 - 25 \)
This is a difference of squares. It can be factored as:
\[
y^2 - 25 = (y - 5)(y + 5)
\]

### 3. \( 9y^2 - 16 \)
This is another difference of squares. The square root of \(9y^2\) is \(3y\) and the square root of \(16\) is \(4\), so:
\[
9y^2 - 16 = (3y - 4)(3y + 4)
\]

### 4. \( 4y^2 - 12y + 9 \)
This is also a perfect square trinomial. We need to check if it can be factored into the form \((ay + b)^2\). Here, \(4y^2\) is a square, and \(9\) is a square. Factoring, we get:
\[
4y^2 - 12y + 9 = (2y - 3)(2y - 3) = (2y - 3)^2
\]

Let me know if you would like further explanation or details!

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**Relative questions:**
1. How do you factor a perfect square trinomial?
2. What is the process for factoring a difference of squares?
3. Can every quadratic expression be factored? If not, why?
4. What are the special cases where factoring becomes easier?
5. How do you recognize a perfect square trinomial quickly?

**Tip**: When factoring, always check for a greatest common factor (GCF) before using any other methods.

Rewrite each of the following polynomial expressions in factored form:
1. y^2 + 8y + 16
2. y^2 - 25
3. 9y^2 - 16
4. 4y^2 - 12y + 9

Learn how to factor polynomial expressions including perfect square trinomials and differences of squares. This problem covers algebraic concepts suitable for students in Grades 8-10, with detailed solutions for each polynomial.

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