Let `~=` be an operation on a and b defined as a `~=` b=ab+`b^(2)`. If p `~=` q=0 where p and q are non-zero integers, which of the following options is true?

Options

A. p+q=1
B. p+2q=0
C. p+3q=2p
D. p-2q= -3q

Answer

To solve the problem, we start with the operation defined as \( a \sim b = ab + b^2 \). We need to find the conditions under which \( p \sim q = 0 \) for non-zero integers \( p \) and \( q \).

### Step-by-Step Solution:

1. **Define the Operation**:
We have the operation defined as:
\[ p \sim q = pq + q^2 \]

2. **Set Up the Equation**:
According to the problem, we need to set this equal to zero:
\[ pq + q^2 = 0 \]

3. **Factor the Equation**:
We can factor out \( q \) from the left-hand side:
\[ q(p + q) = 0 \]

4. **Analyze the Factors**:
Since \( p \) and \( q \) are both non-zero integers, \( q \) cannot be zero. Therefore, we must have:
\[ p + q = 0 \]

5. **Solve for \( p \)**:
From the equation \( p + q = 0 \), we can express \( p \) in terms of \( q \):
\[ p = -q \]

6. **Evaluate the Options**:
Now we need to check which of the given options is true based on \( p + q = 0 \):
- Option A: \( p + q = 1 \) (not true)
- Option B: \( p + 2q = 0 \) (not true)
- Option C: \( p + 3q = p \) (not true)
- Option D: \( p - 2q = 0 \) (this simplifies to \( p = 2q \), which is not true)

However, if we manipulate Option D:
\[ p - 2q = -3q \implies p + q = 0 \]
This is consistent with our derived equation.

### Conclusion:
The correct interpretation leads us to conclude that the valid condition is:
\[ p + q = 0 \]
Thus, the correct option is **D**.