`x^(2)+x-42=0`
`{:("Quantity A","Quantity B"),(|x+1|,5):}`

Options

A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the infromation given.

Answer

To solve the equation \( x^2 + x - 42 = 0 \) and compare the quantities \( |x + 1| \) and \( 5 \), we will follow these steps:

### Step 1: Solve the quadratic equation
The given equation is:
\[ x^2 + x - 42 = 0 \]
We can factor this quadratic equation. We need to find two numbers that multiply to \(-42\) (the product of \(A\) and \(C\)) and add up to \(1\) (the coefficient of \(B\)). The numbers that satisfy this are \(7\) and \(-6\).

### Step 2: Factor the equation
Using the numbers found, we can rewrite the equation:
\[ x^2 + 7x - 6x - 42 = 0 \]
Now, we can group the terms:
\[ (x^2 + 7x) + (-6x - 42) = 0 \]
Factoring by grouping gives us:
\[ x(x + 7) - 6(x + 7) = 0 \]
Factoring out the common term \((x + 7)\):
\[ (x + 7)(x - 6) = 0 \]

### Step 3: Find the roots
Setting each factor to zero gives us:
\[ x + 7 = 0 \quad \Rightarrow \quad x = -7 \]
\[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \]
Thus, the solutions to the equation are \(x = -7\) and \(x = 6\).

### Step 4: Calculate Quantity A
Now we need to evaluate \( |x + 1| \) for both values of \(x\).

1. For \(x = 6\):
\[ |x + 1| = |6 + 1| = |7| = 7 \]

2. For \(x = -7\):
\[ |x + 1| = |-7 + 1| = |-6| = 6 \]

### Step 5: Compare Quantity A and Quantity B
Now we compare the results with Quantity B, which is \(5\):
- When \(x = 6\), \( |x + 1| = 7 \) which is greater than \(5\).
- When \(x = -7\), \( |x + 1| = 6 \) which is also greater than \(5\).

### Conclusion
In both cases, \( |x + 1| > 5 \). Therefore, we conclude that:
\[ \text{Quantity A is greater than Quantity B.} \]

### Final Answer
The correct option is that Quantity A is greater than Quantity B.

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