The factor form `x^(2)+9x+14` is

Options

A. `(x+7)(x+2)`
B. `(x-7)(x+7)`
C. `(2x+7)(3x+3)`
D. `(x+3)(x+4)`

Answer

To factor the quadratic expression \(x^2 + 9x + 14\), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the standard form \(ax^2 + bx + c\), where:
- \(a = 1\) (coefficient of \(x^2\))
- \(b = 9\) (coefficient of \(x\))
- \(c = 14\) (constant term)

### Step 2: Find the product and sum
We need to find two numbers that multiply to \(c\) (which is 14) and add up to \(b\) (which is 9).

- Product = \(c = 14\)
- Sum = \(b = 9\)

### Step 3: List the factor pairs of 14
The pairs of factors of 14 are:
- \(1 \times 14\)
- \(2 \times 7\)

### Step 4: Identify the correct pair
From the factor pairs, we need to find which pair adds up to 9:
- \(1 + 14 = 15\) (not suitable)
- \(2 + 7 = 9\) (suitable)

Thus, the two numbers we are looking for are \(2\) and \(7\).

### Step 5: Rewrite the middle term
Now, we can rewrite the expression by breaking the middle term \(9x\) into \(2x + 7x\):
\[ x^2 + 2x + 7x + 14 \]

### Step 6: Group the terms
Next, we group the terms:
\[ (x^2 + 2x) + (7x + 14) \]

### Step 7: Factor out the common terms
Now, we factor out the common factors from each group:
\[ x(x + 2) + 7(x + 2) \]

### Step 8: Factor by grouping
Now, we can factor out the common binomial factor \((x + 2)\):
\[ (x + 2)(x + 7) \]

### Final Answer
Thus, the factor form of the expression \(x^2 + 9x + 14\) is:
\[ (x + 2)(x + 7) \]