`(x^2-4x-21)=?`

Options

A. `(x-7)(x-3)`
B. `(x+7)(x-3)`
C. `(x-7)(x+3)`
D. None of these

Explanation

To factor the polynomial \( x^2 - 4x - 21 \), 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 \( x^2 \))
- \( b = -4 \) (coefficient of \( x \))
- \( c = -21 \) (constant term)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( ac = 1 \times -21 = -21 \) and add to \( b = -4 \).

The pairs of factors of \(-21\) are:
- \( 1 \) and \(-21\)
- \(-1\) and \(21\)
- \( 3 \) and \(-7\)
- \(-3\) and \(7\)

Among these, the pair \( 3 \) and \(-7\) adds up to \(-4\):
\[ 3 + (-7) = -4 \]

### Step 3: Rewrite the middle term using the two numbers
We can rewrite the polynomial by splitting the middle term:
\[ x^2 - 4x - 21 = x^2 + 3x - 7x - 21 \]

### Step 4: Factor by grouping
Now, we will group the terms:
\[ (x^2 + 3x) + (-7x - 21) \]

Next, we factor out the common factors from each group:
\[ x(x + 3) - 7(x + 3) \]

### Step 5: Factor out the common binomial
Now, we can factor out the common binomial \( (x + 3) \):
\[ (x + 3)(x - 7) \]

### Final Answer
Thus, the factored form of the polynomial \( x^2 - 4x - 21 \) is:
\[ (x + 3)(x - 7) \]

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