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 given polynomial is in the form \( ax^2 + bx + c \), where:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = 9 \) (coefficient of \( x \))
- \( c = 20 \) (constant term)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( ac = 1 \times 20 = 20 \) and add to \( b = 9 \).

### Step 3: List the factor pairs of 20
The pairs of factors of 20 are:
- \( 1 \times 20 \)
- \( 2 \times 10 \)
- \( 4 \times 5 \)

### Step 4: Check which pair adds up to 9
Now we check which of these pairs adds up to 9:
- \( 1 + 20 = 21 \) (not suitable)
- \( 2 + 10 = 12 \) (not suitable)
- \( 4 + 5 = 9 \) (suitable)

Thus, the two numbers we are looking for are \( 4 \) and \( 5 \).

### Step 5: Rewrite the middle term using the two numbers
We can rewrite the polynomial \( x^2 + 9x + 20 \) as:
\[ x^2 + 4x + 5x + 20 \]

### Step 6: Factor by grouping
Now, we group the terms:
\[ (x^2 + 4x) + (5x + 20) \]
Now, we factor out the common factors from each group:
- From \( x^2 + 4x \), we can factor out \( x \):
\[ x(x + 4) \]
- From \( 5x + 20 \), we can factor out \( 5 \):
\[ 5(x + 4) \]

### Step 7: Combine the factors
Now we can combine the two groups:
\[ x(x + 4) + 5(x + 4) \]
This can be factored further as:
\[ (x + 4)(x + 5) \]

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

---