On of the factors of
`4x ^(2) + y ^(2) + 14 x - 7y - 4 xy +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: Rewrite the expression
Start with the given expression:
\[ 4x^2 + y^2 + 14x - 7y - 4xy + 12 \]

### Step 2: Group terms
Group the terms in a way that allows us to factor them more easily:
\[ (4x^2 - 4xy + y^2) + (14x - 7y) + 12 \]

### Step 3: Factor the quadratic in \(x\) and \(y\)
Notice that \(4x^2 - 4xy + y^2\) can be rewritten as:
\[ (2x - y)^2 \]
So we can rewrite the expression as:
\[ (2x - y)^2 + (14x - 7y) + 12 \]

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

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

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

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

### Conclusion
Thus, one of the factors of the original expression \(4x^2 + y^2 + 14x - 7y - 4xy + 12\) is:
\[ (2x - y + 3) \]
or
\[ (2x - y + 4) \]

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