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 expression \(4x^2 + 8x + 3\), we can follow these steps:

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

### Step 2: Calculate the product of \(a\) and \(c\)
We need to calculate \(a \cdot c\):
\[ 4 \cdot 3 = 12 \]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to \(12\) (the product from step 2) and add up to \(8\) (the coefficient \(b\)):
The numbers that satisfy this are \(6\) and \(2\) because:
\[ 6 \cdot 2 = 12 \quad \text{and} \quad 6 + 2 = 8 \]

### Step 4: Rewrite the middle term using the two numbers
Now we can rewrite the expression by splitting the middle term \(8x\) into \(6x + 2x\):
\[ 4x^2 + 6x + 2x + 3 \]

### Step 5: Factor by grouping
Next, we can group the terms:
\[ (4x^2 + 6x) + (2x + 3) \]
Now, factor out the common factors in each group:
\[ 2x(2x + 3) + 1(2x + 3) \]

### Step 6: Factor out the common binomial
Now we can factor out the common binomial \((2x + 3)\):
\[ (2x + 3)(2x + 1) \]

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