`3x^(2)+11x - 4 ` = _______

Options

A. `(3x-1)(x+4)`
B. `(3x-1)(x-4)`
C. `(3x+1)(x-4)`
D. `(3x+1)(x+4)`

Answer

To factorize the expression \(3x^2 + 11x - 4\), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \(ax^2 + bx + c\), where:
- \(a = 3\)
- \(b = 11\)
- \(c = -4\)

### Step 2: Multiply \(a\) and \(c\)
We need to multiply \(a\) and \(c\):
\[ a \cdot c = 3 \cdot (-4) = -12 \]

### Step 3: Find two numbers that multiply to \(a \cdot c\) and add to \(b\)
We need to find two numbers that multiply to \(-12\) and add to \(11\). The numbers that satisfy this condition are \(12\) and \(-1\) because:
\[ 12 \cdot (-1) = -12 \quad \text{and} \quad 12 + (-1) = 11 \]

### Step 4: Rewrite the middle term using the two numbers
Now, we can rewrite the expression by breaking the middle term \(11x\) into \(12x - 1x\):
\[ 3x^2 + 12x - 1x - 4 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (3x^2 + 12x) + (-1x - 4) \]

### Step 6: Factor by grouping
Now, we factor out the common factors from each group:
- From the first group \(3x^2 + 12x\), we can factor out \(3x\):
\[ 3x(x + 4) \]
- From the second group \(-1x - 4\), we can factor out \(-1\):
\[ -1(x + 4) \]

### Step 7: Combine the factors
Now we can combine the factored groups:
\[ 3x(x + 4) - 1(x + 4) = (3x - 1)(x + 4) \]

### Final Result
Thus, the factorization of the expression \(3x^2 + 11x - 4\) is:
\[ (3x - 1)(x + 4) \]

---