`(4x^2+4x-3)=?`

Options

A. `(2x-1)(2x-3)`
B. `(2x+1)(2x-3)`
C. `(2x+3)(2x-1)`
D. None of these

Explanation

To factor the polynomial \(4x^2 + 4x - 3\), we can follow these steps:

### Step 1: Identify the coefficients
The polynomial can be expressed in the standard form \(ax^2 + bx + c\), where:
- \(a = 4\)
- \(b = 4\)
- \(c = -3\)

### Step 2: Calculate the product \(ac\)
We need to find the product of \(a\) and \(c\):
\[ ac = 4 \times (-3) = -12 \]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to \(-12\) (the value of \(ac\)) and add up to \(4\) (the value of \(b\)). The numbers that satisfy these conditions are \(6\) and \(-2\):
\[ 6 \times (-2) = -12 \quad \text{and} \quad 6 + (-2) = 4 \]

### Step 4: Rewrite the middle term
Using the numbers found, we can rewrite the polynomial:
\[ 4x^2 + 6x - 2x - 3 \]

### Step 5: Group the terms
Now, we will group the terms into two pairs:
\[ (4x^2 + 6x) + (-2x - 3) \]

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

### Step 7: Combine the factored groups
Now we can combine the factored terms:
\[ 2x(2x + 3) - 1(2x + 3) \]
This can be factored further:
\[ (2x + 3)(2x - 1) \]

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

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