Let's determine which one of the given trinomials is not a perfect square trinomial by analyzing each of them individually.
### Step 1: Understand what a perfect square trinomial is
A perfect square trinomial is a quadratic trinomial of the form [tex]\( (ax + b)^2 \)[/tex] which expands to:
[tex]\[
a^2 x^2 + 2abx + b^2
\][/tex]
We need to check if each of the provided trinomials match this form.
### Step 2: Analyze each trinomial
#### Trinomial 1: [tex]\( x^2 - 16x + 64 \)[/tex]
To check if this is a perfect square trinomial:
1. Identify the first term [tex]\( x^2 \)[/tex] which is already a perfect square ([tex]\((x)^2 \)[/tex]).
2. Identify the last term 64 which is a perfect square ([tex]\((8)^2 \)[/tex]).
3. Check the middle term: [tex]\( 2 \cdot x \cdot 8 = 16x \)[/tex] matches the middle term [tex]\( -16x \)[/tex] if [tex]\((x-8)^2\)[/tex].
Therefore, [tex]\( x^2 - 16x + 64 = (x - 8)^2 \)[/tex]. This is a perfect square trinomial.
#### Trinomial 2: [tex]\( x^2 - 12x + 16 \)[/tex]
To check if this is a perfect square trinomial:
1. Identify the first term [tex]\( x^2 \)[/tex] which is already a perfect square ([tex]\((x)^2 \)[/tex]).
2. Identify the last term 16 which is a perfect square ([tex]\((4)^2 \)[/tex]).
3. Check the middle term: [tex]\( 2 \cdot x \cdot 4 = 8x \)[/tex] does not match the middle term [tex]\( -12x \)[/tex].
Therefore, [tex]\( x^2 - 12x + 16 \)[/tex] is not a perfect square trinomial.
#### Trinomial 3: [tex]\( x^2 - 6x + 9 \)[/tex]
To check if this is a perfect square trinomial:
1. Identify the first term [tex]\( x^2 \)[/tex] which is already a perfect square ([tex]\((x)^2 \)[/tex]).
2. Identify the last term 9 which is a perfect square ([tex]\((3)^2 \)[/tex]).
3. Check the middle term: [tex]\( 2 \cdot x \cdot 3 = 6x \)[/tex] matches the middle term [tex]\( -6x \)[/tex] if [tex]\((x-3)^2\)[/tex].
Therefore, [tex]\( x^2 - 6x + 9 = (x - 3)^2 \)[/tex]. This is a perfect square trinomial.
#### Trinomial 4: [tex]\( x^2 + 6x + 9 \)[/tex]
To check if this is a perfect square trinomial:
1. Identify the first term [tex]\( x^2 \)[/tex] which is already a perfect square ([tex]\((x)^2 \)[/tex]).
2. Identify the last term 9 which is a perfect square ([tex]\((3)^2 \)[/tex]).
3. Check the middle term: [tex]\( 2 \cdot x \cdot 3 = 6x \)[/tex] matches the middle term [tex]\( 6x \)[/tex] if [tex]\((x+3)^2\)[/tex].
Therefore, [tex]\( x^2 + 6x + 9 = (x + 3)^2 \)[/tex]. This is a perfect square trinomial.
### Conclusion
The only trinomial that is not a perfect square trinomial is:
[tex]\[
\boxed{x^2 - 12x + 16}
\][/tex]
