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 can follow these steps:

### Step 1: Identify the coefficients
The given 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 \(c\):
\[ a \cdot c = 4 \cdot 5 = 20 \]

### Step 3: Find two numbers that multiply to \(20\) and add to \(12\)
We need to find two numbers \(m\) and \(n\) such that:
- \(m \cdot n = 20\)
- \(m + n = 12\)

The pairs of factors of \(20\) are:
- \(1 \cdot 20\)
- \(2 \cdot 10\)
- \(4 \cdot 5\)

Among these, the pair \(2\) and \(10\) adds up to \(12\):
\[ 2 + 10 = 12 \]

### Step 4: Rewrite the middle term using the numbers found
Now, we can rewrite the polynomial:
\[ 4x^2 + 10x + 2x + 5 \]

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

Now, we factor out the common factors from each group:
1. From \(4x^2 + 10x\), we can factor out \(2x\):
\[ 2x(2x + 5) \]
2. From \(2x + 5\), we can factor out \(1\):
\[ 1(2x + 5) \]

Now, we can combine these:
\[ 2x(2x + 5) + 1(2x + 5) = (2x + 5)(2x + 1) \]

### Step 6: Write the final factorised form
Thus, the factorised form of the polynomial \(4x^2 + 12x + 5\) is:
\[ (2x + 5)(2x + 1) \]

### Summary
The final answer is:
\[ 4x^2 + 12x + 5 = (2x + 5)(2x + 1) \]

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