an example of an equality relation of two expressions in x, which of not an identity is

Options

A. `( x + 3) ^(2) = x^(2) + 6 x + 9 `
B. ` ( x + 2 y) ^(3) = x^(3) + 8 y^(3) + 6xy ( x + 2 y)`
C. ` ( x + 2)^(2) = x^(2) + 2 x + 4 `
D. ` ( x + 3) ( x - 3) = x^(2) - 9 `

Answer

To solve the problem, we need to find an example of an equality relation of two expressions in \( x \) that is not an identity. An identity is an equation that is true for all values of the variable.

Let's analyze the options given in the video transcript step by step.

### Step 1: Identify the expressions
We have four options to consider. We need to evaluate each expression to determine if it is an identity or not.

### Step 2: Evaluate the first option
**Option 1:** \( (x + 3)^3 = x^3 + 27 \)

Using the binomial expansion:
\[ (x + 3)^3 = x^3 + 3 \cdot x^2 \cdot 3 + 3 \cdot x \cdot 3^2 + 3^3 \]
\[ = x^3 + 9x^2 + 27x + 27 \]
This is not equal to \( x^3 + 27 \). Therefore, this is not an identity.

### Step 3: Evaluate the second option
**Option 2:** \( (x + 2y)^3 = x^3 + 8y^3 + 6xy(x + 2y) \)

Using the binomial expansion:
\[ (x + 2y)^3 = x^3 + 3 \cdot x^2 \cdot 2y + 3 \cdot x \cdot (2y)^2 + (2y)^3 \]
\[ = x^3 + 6xy + 12y^2 + 8y^3 \]
This does not simplify to \( x^3 + 8y^3 + 6xy(x + 2y) \). Therefore, this is also not an identity.

### Step 4: Evaluate the third option
**Option 3:** \( (x + 2)^2 = x^2 + 4 \)

Using the binomial expansion:
\[ (x + 2)^2 = x^2 + 2 \cdot x \cdot 2 + 2^2 \]
\[ = x^2 + 4x + 4 \]
This is not equal to \( x^2 + 4 \). Therefore, this is not an identity.

### Step 5: Evaluate the fourth option
**Option 4:** \( (x + 3)(x - 3) = x^2 - 9 \)

Using the difference of squares:
\[ (x + 3)(x - 3) = x^2 - 9 \]
This is true for all \( x \), hence this is an identity.

### Conclusion
From the evaluations, we can conclude that the expressions in options 1, 2, and 3 are not identities, while option 4 is an identity.

### Final Answer
The example of an equality relation of two expressions in \( x \) which is not an identity is:
- \( (x + 3)^3 = x^3 + 27 \)