Factor the perfect square trinomials

To factor the perfect square trinomials, we use the formula:

\[
a^2 + 2ab + b^2 = (a + b)^2
\]

Now, let's factor the given trinomials:

1) \( x^2 + 6x + 9 \)

Here, \( a = x \) and \( b = 3 \) (because \( 3^2 = 9 \) and \( 2ab = 6x \) implies \( 2 \cdot x \cdot 3 = 6x \)). Thus, we have:
\[
x^2 + 6x + 9 = (x + 3)^2
\]

2) \( x^2 + 10x + 25 \)

Here, \( a = x \) and \( b = 5 \) (because \( 5^2 = 25 \) and \( 2ab = 10x \)). Thus, we have:
\[
x^2 + 10x + 25 = (x + 5)^2
\]

3) \( 9x^2 + 12x + 4 \)

Here, \( a = 3x \) and \( b = 2 \) (because \( 2^2 = 4 \) and \( 2ab = 12x \) implies \( 2 \cdot 3x \cdot 2 = 12x \)). Thus, we have:
\[
9x^2 + 12x + 4 = (3x + 2)^2
\]

4) \( 9x^2 - 30x + 25 \)

Here, \( a = 3x \) and \( b = 5 \) (because \( 5^2 = 25 \) and \( 2ab = -30x \)). Thus, we have:
\[
9x^2 - 30x + 25 = (3x - 5)^2
\]

So the factorizations are:

1) \( (x + 3)^2 \)

2) \( (x + 5)^2 \)

3) \( (3x + 2)^2 \)

4) \( (3x - 5)^2 \)

does the answer have to be double quotationed?

No, the answers do not need to be double-quoted. You can present the factorizations without quotes. Here are the factorizations again without quotes:

1) \( (x + 3)^2 \)

2) \( (x + 5)^2 \)

3) \( (3x + 2)^2 \)

4) \( (3x - 5)^2 \)

Feel free to format them as you prefer!