Chart displaying examples of perfect square trinomials and non-perfect squares for algebraic identification.

To solve the problem of identifying which trinomials are perfect squares and which are not, we need to use the definition of a perfect square trinomial. A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. The general forms of perfect square trinomials are:

1. \( (a + b)^2 = a^2 + 2ab + b^2 \)
2. \( (a - b)^2 = a^2 - 2ab + b^2 \)

Steps to Identify Perfect Square Trinomials:

1. Check the first and last terms: They should be perfect squares.
2. Check the middle term: It should be twice the product of the square roots of the first and last terms.

Let's analyze each trinomial in the table step by step.

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Perfect Square Trinomials

#### 1. \( x^2 + 2x + 1 \)
- First term: \( x^2 \) (perfect square of \( x \))
- Last term: \( 1 \) (perfect square of \( 1 \))
- Middle term: \( 2x \) (twice the product of \( x \) and \( 1 \))
- This matches the form \( (x + 1)^2 \).

#### 2. \( x^2 + 8x + 16 \)
- First term: \( x^2 \) (perfect square of \( x \))
- Last term: \( 16 \) (perfect square of \( 4 \))
- Middle term: \( 8x \) (twice the product of \( x \) and \( 4 \))
- This matches the form \( (x + 4)^2 \).

#### 3. \( x^2 - 8x + 64 \)
- First term: \( x^2 \) (perfect square of \( x \))
- Last term: \( 64 \) (perfect square of \( 8 \))
- Middle term: \( -8x \) (twice the product of \( x \) and \( -8 \))
- This matches the form \( (x - 8)^2 \).

#### 4. \( x^2 - 4x + 4 \)
- First term: \( x^2 \) (perfect square of \( x \))
- Last term: \( 4 \) (perfect square of \( 2 \))
- Middle term: \( -4x \) (twice the product of \( x \) and \( -2 \))
- This matches the form \( (x - 2)^2 \).

#### 5. \( x^2 + 12x + 36 \)
- First term: \( x^2 \) (perfect square of \( x \))
- Last term: \( 36 \) (perfect square of \( 6 \))
- Middle term: \( 12x \) (twice the product of \( x \) and \( 6 \))
- This matches the form \( (x + 6)^2 \).

#### 6. \( 9x^2 - 12x + 4 \)
- First term: \( 9x^2 \) (perfect square of \( 3x \))
- Last term: \( 4 \) (perfect square of \( 2 \))
- Middle term: \( -12x \) (twice the product of \( 3x \) and \( -2 \))
- This matches the form \( (3x - 2)^2 \).

#### 7. \( x^2 - 10x + 100 \)
- First term: \( x^2 \) (perfect square of \( x \))
- Last term: \( 100 \) (perfect square of \( 10 \))
- Middle term: \( -10x \) (twice the product of \( x \) and \( -10 \))
- This matches the form \( (x - 10)^2 \).

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Not Perfect Square Trinomials

#### 1. \( x^2 + 2x + 2 \)
- First term: \( x^2 \) (perfect square of \( x \))
- Last term: \( 2 \) (not a perfect square)
- This cannot be a perfect square trinomial.

#### 2. \( 4x^2 + 4x + 4 \)
- First term: \( 4x^2 \) (perfect square of \( 2x \))
- Last term: \( 4 \) (perfect square of \( 2 \))
- Middle term: \( 4x \) (not twice the product of \( 2x \) and \( 2 \))
- This does not match the form of a perfect square trinomial.

#### 3. \( x^2 - 4x - 4 \)
- First term: \( x^2 \) (perfect square of \( x \))
- Last term: \( -4 \) (not a perfect square)
- This cannot be a perfect square trinomial.

#### 4. \( x^2 - 16x - 64 \)
- First term: \( x^2 \) (perfect square of \( x \))
- Last term: \( -64 \) (not a perfect square)
- This cannot be a perfect square trinomial.

#### 5. \( x^2 + 16x + 64 \)
- First term: \( x^2 \) (perfect square of \( x \))
- Last term: \( 64 \) (perfect square of \( 8 \))
- Middle term: \( 16x \) (twice the product of \( x \) and \( 8 \))
- This matches the form \( (x + 8)^2 \). (This was incorrectly labeled as "Not Perfect Squares.")

#### 6. \( x^2 + 20x + 100 \)
- First term: \( x^2 \) (perfect square of \( x \))
- Last term: \( 100 \) (perfect square of \( 10 \))
- Middle term: \( 20x \) (twice the product of \( x \) and \( 10 \))
- This matches the form \( (x + 10)^2 \). (This was incorrectly labeled as "Not Perfect Squares.")

#### 7. \( 25x^2 + 60x + 36 \)
- First term: \( 25x^2 \) (perfect square of \( 5x \))
- Last term: \( 36 \) (perfect square of \( 6 \))
- Middle term: \( 60x \) (twice the product of \( 5x \) and \( 6 \))
- This matches the form \( (5x + 6)^2 \). (This was incorrectly labeled as "Not Perfect Squares.")

#### 8. \( x^2 - 14x + 49 \)
- First term: \( x^2 \) (perfect square of \( x \))
- Last term: \( 49 \) (perfect square of \( 7 \))
- Middle term: \( -14x \) (twice the product of \( x \) and \( -7 \))
- This matches the form \( (x - 7)^2 \). (This was incorrectly labeled as "Not Perfect Squares.")

#### 9. \( x^2 + 16x + 16 \)
- First term: \( x^2 \) (perfect square of \( x \))
- Last term: \( 16 \) (perfect square of \( 4 \))
- Middle term: \( 16x \) (not twice the product of \( x \) and \( 4 \))
- This does not match the form of a perfect square trinomial.

#### 10. \( x^2 - 81 \)
- First term: \( x^2 \) (perfect square of \( x \))
- Last term: \( -81 \) (not a perfect square)
- This cannot be a perfect square trinomial.

#### 11. \( 4x^2 + 4x + 1 \)
- First term: \( 4x^2 \) (perfect square of \( 2x \))
- Last term: \( 1 \) (perfect square of \( 1 \))
- Middle term: \( 4x \) (twice the product of \( 2x \) and \( 1 \))
- This matches the form \( (2x + 1)^2 \). (This was incorrectly labeled as "Not Perfect Squares.")

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Corrected Table:

After re-evaluating, the corrected classification is:

| Perfect Square Trinomials | Not Perfect Square Trinomials |
|---------------------------|-------------------------------|
| \( x^2 + 2x + 1 \) | \( x^2 + 2x + 2 \) |
| \( x^2 + 8x + 16 \) | \( 4x^2 + 4x + 4 \) |
| \( x^2 - 8x + 64 \) | \( x^2 - 4x - 4 \) |
| \( x^2 - 4x + 4 \) | \( x^2 - 16x - 64 \) |
| \( x^2 + 12x + 36 \) | \( x^2 + 20x + 100 \) |
| \( 9x^2 - 12x + 4 \) | \( 25x^2 + 60x + 36 \) |
| \( x^2 - 10x + 100 \) | \( x^2 - 14x + 49 \) |
| | \( x^2 + 16x + 16 \) |
| | \( x^2 - 81 \) |
| | \( 4x^2 + 4x + 1 \) |

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Final Answer:

\[
\boxed{
\begin{array}{|c|c|}
\hline
\text{Perfect Square Trinomials} & \text{Not Perfect Square Trinomials} \\
\hline
x^2 + 2x + 1 & x^2 + 2x + 2 \\
x^2 + 8x + 16 & 4x^2 + 4x + 4 \\
x^2 - 8x + 64 & x^2 - 4x - 4 \\
x^2 - 4x + 4 & x^2 - 16x - 64 \\
x^2 + 12x + 36 & x^2 + 20x + 100 \\
9x^2 - 12x + 4 & 25x^2 + 60x + 36 \\
x^2 - 10x + 100 & x^2 - 14x + 49 \\
& x^2 + 16x + 16 \\
& x^2 - 81 \\
& 4x^2 + 4x + 1 \\
\hline
\end{array}
}
\]