`x^(2)+11x+24 =` ________

Options

A. `(x+6)(x+4)`
B. `(x+12)(x-20)`
C. `(x+12)(x+2)`
D. `(x+8)(x+3)`

Answer

To factor the quadratic expression \(x^2 + 11x + 24\), we will follow these steps:

### Step 1: Identify the coefficients
We start with the quadratic expression:
\[ x^2 + 11x + 24 \]
Here, the coefficient of \(x^2\) (which we denote as \(a\)) is 1, the coefficient of \(x\) (denote as \(b\)) is 11, and the constant term (denote as \(c\)) is 24.

### Step 2: Find two numbers that multiply and add
We need to find two numbers that:
- Multiply to \(c\) (which is 24)
- Add up to \(b\) (which is 11)

### Step 3: List the factor pairs of 24
The factor pairs of 24 are:
- \(1 \times 24\)
- \(2 \times 12\)
- \(3 \times 8\)
- \(4 \times 6\)

### Step 4: Check which pair adds up to 11
Now, we check which of these pairs adds up to 11:
- \(1 + 24 = 25\)
- \(2 + 12 = 14\)
- \(3 + 8 = 11\) (This is the pair we need!)
- \(4 + 6 = 10\)

The correct pair is \(3\) and \(8\).

### Step 5: Write the factors
Since the numbers \(3\) and \(8\) add up to \(11\) and multiply to \(24\), we can write the factored form of the quadratic expression:
\[ x^2 + 11x + 24 = (x + 3)(x + 8) \]

### Final Answer
Thus, the factored form of \(x^2 + 11x + 24\) is:
\[ (x + 3)(x + 8) \]

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