`xyzlt0`
`{:("Quantity A","Quantity B"),(x+y+z,2x+2y+2z):}`

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 problem, we need to compare two quantities given the condition that the product \(xyz < 0\). This means that either one of the numbers \(x\), \(y\), or \(z\) is negative while the other two are positive, or all three numbers are negative.

### Step-by-Step Solution:

1. **Understanding the Condition**:
Since \(xyz < 0\), we have two scenarios:
- **Scenario 1**: One number is negative and the other two are positive.
- **Scenario 2**: All three numbers are negative.

2. **Calculating Quantity A and Quantity B**:
- **Quantity A**: \(x + y + z\)
- **Quantity B**: \(2x + 2y + 2z = 2(x + y + z)\)

3. **Analyzing Scenario 1**:
- Let’s take an example where \(x = -1\), \(y = 2\), and \(z = 3\).
- Calculate Quantity A:
\[ A = x + y + z = -1 + 2 + 3 = 4 \]
- Calculate Quantity B:
\[ B = 2(x + y + z) = 2 \times 4 = 8 \]
- Here, \(B > A\).

4. **Analyzing Scenario 2**:
- Now, let’s take an example where \(x = -1\), \(y = -2\), and \(z = -3\).
- Calculate Quantity A:
\[ A = x + y + z = -1 - 2 - 3 = -6 \]
- Calculate Quantity B:
\[ B = 2(x + y + z) = 2 \times (-6) = -12 \]
- Here, \(A > B\).

5. **Conclusion**:
- From the two scenarios, we see that in the first case, Quantity B is greater, while in the second case, Quantity A is greater.
- Therefore, there is no consistent relationship between Quantity A and Quantity B based on the given condition \(xyz < 0\).

### Final Answer:
The relationship between Quantity A and Quantity B cannot be determined from the information given. Thus, the correct option is **Option 4**.