Factorise `4x^(2)+12x+5`.

Options

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

Explanation

To factorise the polynomial \(4x^2 + 12x + 5\), we will follow these steps:

### Step 1: Identify the coefficients
The polynomial is in the form \(ax^2 + bx + c\), where:
- \(a = 4\)
- \(b = 12\)
- \(c = 5\)

### Step 2: Multiply \(a\) and \(c\)
We need to multiply \(a\) and \(c\):
\[ ac = 4 \times 5 = 20 \]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to \(20\) (the value of \(ac\)) and add to \(12\) (the value of \(b\)). The numbers that satisfy this are \(10\) and \(2\) because:
\[ 10 \times 2 = 20 \quad \text{and} \quad 10 + 2 = 12 \]

### Step 4: Rewrite the middle term
We can rewrite the polynomial by breaking the middle term \(12x\) into \(10x\) and \(2x\):
\[ 4x^2 + 10x + 2x + 5 \]

### Step 5: Group the terms
Now, we will group the terms:
\[ (4x^2 + 10x) + (2x + 5) \]

### Step 6: Factor out the common factors from each group
From the first group \(4x^2 + 10x\), we can factor out \(2x\):
\[ 2x(2x + 5) \]
From the second group \(2x + 5\), we can factor out \(1\):
\[ 1(2x + 5) \]

### Step 7: Combine the factors
Now we can combine the factored groups:
\[ 2x(2x + 5) + 1(2x + 5) = (2x + 5)(2x + 1) \]

### Final Result
Thus, the factorization of the polynomial \(4x^2 + 12x + 5\) is:
\[ (2x + 5)(2x + 1) \]

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