Factorise `x^(2)+9x+20`.

Options

A. `(x+4)(x+4)`
B. `(x+5)(x+4)`
C. `(x+5)(x-4)`
D. `(x-5)(x+4)`

Explanation

To factorize the polynomial \(x^2 + 9x + 20\), we will follow these steps:

### Step 1: Identify the coefficients
The polynomial is in the standard form \(ax^2 + bx + c\). Here, we have:
- \(a = 1\) (coefficient of \(x^2\))
- \(b = 9\) (coefficient of \(x\))
- \(c = 20\) (constant term)

### Step 2: Find two numbers that multiply to \(c\) and add to \(b\)
We need to find two numbers that:
- Multiply to \(c = 20\)
- Add up to \(b = 9\)

Let's list the pairs of factors of 20:
- \(1 \times 20\)
- \(2 \times 10\)
- \(4 \times 5\)

Now, we check which pair adds up to 9:
- \(1 + 20 = 21\) (not a match)
- \(2 + 10 = 12\) (not a match)
- \(4 + 5 = 9\) (this is a match)

### Step 3: Rewrite the middle term using the two numbers
Now that we have found the numbers \(4\) and \(5\), we can rewrite the polynomial:
\[ x^2 + 4x + 5x + 20 \]

### Step 4: Group the terms
Next, we group the terms:
\[ (x^2 + 4x) + (5x + 20) \]

### Step 5: Factor out the common factors from each group
Now, we factor out the common factors from each group:
\[ x(x + 4) + 5(x + 4) \]

### Step 6: Factor out the common binomial
Now we can see that \((x + 4)\) is a common factor:
\[ (x + 4)(x + 5) \]

### Final Answer
Thus, the factorization of the polynomial \(x^2 + 9x + 20\) is:
\[ (x + 4)(x + 5) \]

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