The absolte difference between two linear factors of `x ^(2) + 4 xy + 4y ^(2) + x + 2y` is :

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find the absolute difference between two linear factors of the expression \( x^2 + 4xy + 4y^2 + x + 2y \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ x^2 + 4xy + 4y^2 + x + 2y \]

### Step 2: Group the terms
We can group the first three terms and the last two terms:
\[ (x^2 + 4xy + 4y^2) + (x + 2y) \]

### Step 3: Factor the first group
Notice that the first group \( x^2 + 4xy + 4y^2 \) can be factored as a perfect square:
\[ x^2 + 4xy + 4y^2 = (x + 2y)^2 \]

### Step 4: Rewrite the expression with the factor
Now we can rewrite the entire expression as:
\[ (x + 2y)^2 + (x + 2y) \]

### Step 5: Factor out the common term
Let \( z = x + 2y \). Then we can rewrite the expression as:
\[ z^2 + z \]
This can be factored further:
\[ z(z + 1) \]

### Step 6: Substitute back
Substituting back \( z = x + 2y \), we have:
\[ (x + 2y)((x + 2y) + 1) = (x + 2y)(x + 2y + 1) \]

### Step 7: Identify the linear factors
The two linear factors are:
1. \( x + 2y \)
2. \( x + 2y + 1 \)

### Step 8: Find the absolute difference
The absolute difference between these two linear factors is:
\[ |(x + 2y + 1) - (x + 2y)| = |1| = 1 \]

### Final Answer
Thus, the absolute difference between the two linear factors is:
\[ \boxed{1} \]

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