the factorisation of `4x^(2)+8x+3` is

Options

A. `(x+1)(x+3)`
B. `(2x+1)(2x+3)`
C. `(2x+2)(2x+5)`
D. `(2x-1)(2x-3)`

Explanation

To factor the polynomial \(4x^2 + 8x + 3\), we can follow these steps:

### Step 1: Identify the coefficients
The polynomial is in the form \(ax^2 + bx + c\), where:
- \(a = 4\)
- \(b = 8\)
- \(c = 3\)

### 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 add up to \(b\) and multiply to \(ac\)
We need to find two numbers that add up to \(b = 8\) and multiply to \(ac = 12\). The numbers are \(6\) and \(2\) because:
\[ 6 + 2 = 8 \quad \text{and} \quad 6 \times 2 = 12 \]

### Step 4: Rewrite the middle term
We can now rewrite the polynomial by splitting the middle term \(8x\) into \(6x + 2x\):
\[ 4x^2 + 6x + 2x + 3 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (4x^2 + 6x) + (2x + 3) \]

### Step 6: Factor out the common terms in each group
Now, we factor out the common factors from 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 factors
Now we can combine the factored terms:
\[ 2x(2x + 3) + 1(2x + 3) = (2x + 1)(2x + 3) \]

### Final Answer
Thus, the factorization of \(4x^2 + 8x + 3\) is:
\[ (2x + 1)(2x + 3) \]