One of the factors of `4x^(2) + y^(2) + 14x-7y-4xy+12` is equal to

Options

A. `2x+y+4`
B. `2x-y-3`
C. `2x+y-4`
D. `2x-y+3`

Answer

To find one of the factors of the expression \(4x^2 + y^2 + 14x - 7y - 4xy + 12\), we can follow these steps:

### Step 1: Rearrange the expression
We start with the expression:
\[ 4x^2 + y^2 - 4xy + 14x - 7y + 12 \]

### Step 2: Group the terms
We can group the terms in a way that allows us to recognize patterns:
\[ (4x^2 - 4xy + y^2) + (14x - 7y) + 12 \]

### Step 3: Recognize a perfect square
Notice that \(4x^2 - 4xy + y^2\) can be rewritten as:
\[ (2x - y)^2 \]
Thus, we can rewrite the expression as:
\[ (2x - y)^2 + 14x - 7y + 12 \]

### Step 4: Factor out common terms
Now, we can factor out the common terms from \(14x - 7y\):
\[ (2x - y)^2 + 7(2x - y) + 12 \]

### Step 5: Substitute for simplification
Let \(t = 2x - y\). Then the expression becomes:
\[ t^2 + 7t + 12 \]

### Step 6: Factor the quadratic
Now we need to factor \(t^2 + 7t + 12\):
\[ t^2 + 3t + 4t + 12 = t(t + 3) + 4(t + 3) = (t + 3)(t + 4) \]

### Step 7: Substitute back for \(t\)
Now substituting back \(t = 2x - y\):
\[ (2x - y + 3)(2x - y + 4) \]

### Step 8: Identify the factors
Thus, the factors of the original expression are:
\[ (2x - y + 3) \quad \text{and} \quad (2x - y + 4) \]

### Conclusion
One of the factors of the expression \(4x^2 + y^2 + 14x - 7y - 4xy + 12\) is:
\[ 2x - y + 4 \]