If `x^(2)+4xy+4y^(2)+4x+cy+3` can be written as the product of two linear factors then c=

Options

A. -8
B. -2
C. 2
D. 8

Answer

To determine the value of \( c \) such that the expression \( x^2 + 4xy + 4y^2 + 4x + cy + 3 \) can be factored into two linear factors, we will follow these steps:

### Step 1: Identify the coefficients
The given expression is:
\[ x^2 + 4xy + 4y^2 + 4x + cy + 3 \]
We can compare this with the general form of a quadratic equation:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]
From this comparison, we identify the coefficients:
- \( a = 1 \)
- \( h = 2 \)
- \( b = 4 \)
- \( g = 2 \)
- \( f = \frac{c}{2} \)
- The constant term is \( 3 \).

### Step 2: Apply the condition for factorization
For the quadratic to be factored into linear factors, the discriminant \( \Delta \) must be equal to zero. The discriminant is given by the formula:
\[ \Delta = abc + 2fgh - af^2 - bg^2 - ch^2 \]
Substituting the identified values into the discriminant:
- \( a = 1 \)
- \( b = 4 \)
- \( c = 3 \)
- \( g = 2 \)
- \( h = 2 \)
- \( f = \frac{c}{2} \)

We substitute these into the discriminant formula:
\[ \Delta = 1 \cdot 4 \cdot 3 + 2 \cdot \left(\frac{c}{2}\right) \cdot 2 \cdot 2 - 1 \cdot \left(\frac{c}{2}\right)^2 - 4 \cdot 2^2 - 3 \cdot 2^2 \]

### Step 3: Simplify the discriminant
Calculating each term:
1. \( 1 \cdot 4 \cdot 3 = 12 \)
2. \( 2 \cdot \left(\frac{c}{2}\right) \cdot 2 \cdot 2 = 4c \)
3. \( -1 \cdot \left(\frac{c}{2}\right)^2 = -\frac{c^2}{4} \)
4. \( -4 \cdot 4 = -16 \)
5. \( -3 \cdot 4 = -12 \)

Putting it all together:
\[ \Delta = 12 + 4c - \frac{c^2}{4} - 16 - 12 = 0 \]
This simplifies to:
\[ 4c - \frac{c^2}{4} - 16 = 0 \]

### Step 4: Multiply through by 4 to eliminate the fraction
Multiplying the entire equation by 4 gives:
\[ 16c - c^2 - 64 = 0 \]
Rearranging this gives:
\[ c^2 - 16c + 64 = 0 \]

### Step 5: Factor the quadratic
This can be factored as:
\[ (c - 8)^2 = 0 \]
Thus, we find:
\[ c - 8 = 0 \implies c = 8 \]

### Conclusion
The value of \( c \) such that the expression can be factored into two linear factors is:
\[ \boxed{8} \]