Find the facors of `(x^(2) + x - 42)`
A. (x + 14) ( x - 3)
B. ( x + 6) ( x - 7)
B. ( x - 6 ) ( x + 7)
D. ( x - 14) ( x + 3)

Options

A. C
B. B
C. A
D. D

Answer

To find the factors of the quadratic expression \(x^2 + x - 42\), we can follow these steps:

### Step 1: Identify the quadratic expression
We start with the expression:
\[ x^2 + x - 42 \]

### Step 2: Find two numbers that multiply to -42 and add to 1
We need to find two numbers that:
- Multiply to \(-42\) (the constant term)
- Add to \(1\) (the coefficient of \(x\))

After checking the pairs of factors of \(-42\), we find that \(7\) and \(-6\) satisfy these conditions:
\[ 7 \times (-6) = -42 \quad \text{and} \quad 7 + (-6) = 1 \]

### Step 3: Rewrite the middle term using the two numbers
We can rewrite the expression by splitting the middle term:
\[ x^2 + 7x - 6x - 42 \]

### Step 4: Group the terms
Now, we group the terms:
\[ (x^2 + 7x) + (-6x - 42) \]

### Step 5: Factor by grouping
Now, we factor out the common factors from each group:
\[ x(x + 7) - 6(x + 7) \]

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

### Conclusion
Thus, the factors of the expression \(x^2 + x - 42\) are:
\[ (x + 7)(x - 6) \]

### Final Answer
The correct option from the choices given is:
**B. (x + 6)(x - 7)** (Note: This appears to be a mistake in the options as the correct factors are \((x + 7)(x - 6)\)).