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

Options

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

Answer

To factor the quadratic expression \( x^2 + 4x - 21 \), we will follow these steps:

### Step 1: Identify the coefficients
The given expression is \( x^2 + 4x - 21 \). Here, the coefficients are:
- Coefficient of \( x^2 \) (a) = 1
- Coefficient of \( x \) (b) = 4
- Constant term (c) = -21

### Step 2: Multiply the coefficient of \( x^2 \) by the constant term
We multiply the coefficient of \( x^2 \) (which is 1) by the constant term (-21):
\[ 1 \times (-21) = -21 \]

### Step 3: Find two numbers that multiply to -21 and add to 4
We need to find two numbers that multiply to -21 and add to 4. The pairs of factors of -21 are:
- \( 1 \) and \( -21 \)
- \( -1 \) and \( 21 \)
- \( 3 \) and \( -7 \)
- \( -3 \) and \( 7 \)

Out of these pairs, \( 7 \) and \( -3 \) multiply to -21 and add up to 4:
\[ 7 + (-3) = 4 \]

### Step 4: Rewrite the middle term using the two numbers
We can rewrite the expression \( x^2 + 4x - 21 \) as:
\[ x^2 + 7x - 3x - 21 \]

### Step 5: Factor by grouping
Now, we group the terms:
\[ (x^2 + 7x) + (-3x - 21) \]

Next, we factor out the common factors from each group:
- From \( x^2 + 7x \), we can factor out \( x \):
\[ x(x + 7) \]
- From \( -3x - 21 \), we can factor out \( -3 \):
\[ -3(x + 7) \]

Now, we can combine these:
\[ x(x + 7) - 3(x + 7) \]

### Step 6: Factor out the common binomial
Now we can factor out the common binomial \( (x + 7) \):
\[ (x + 7)(x - 3) \]

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

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