Suppose a, b are integers and a + b is a root of `x^2 + ax + b = 0`. What is the maximum possible values of `b^2` ?

Answer

To solve the problem, we need to find the maximum possible value of \( b^2 \) given that \( a + b \) is a root of the quadratic equation \( x^2 + ax + b = 0 \).

### Step-by-Step Solution:

1. **Understanding the Root Condition**:
Since \( a + b \) is a root of the equation \( x^2 + ax + b = 0 \), we can substitute \( x = a + b \) into the equation:
\[ (a + b)^2 + a(a + b) + b = 0 \]

2. **Expanding the Equation**:
Expanding the left-hand side:
\[ (a + b)^2 = a^2 + 2ab + b^2 \]
\[ a(a + b) = a^2 + ab \]
Therefore, substituting back into the equation gives:
\[ a^2 + 2ab + b^2 + a^2 + ab + b = 0 \]
Combining like terms:
\[ 2a^2 + 3ab + b^2 + b = 0 \]

3. **Rearranging the Equation**:
Rearranging the equation, we can express it in terms of \( b \):
\[ b^2 + (3a + 1)b + 2a^2 = 0 \]

4. **Using the Quadratic Formula**:
We can apply the quadratic formula to find \( b \):
\[ b = \frac{-(3a + 1) \pm \sqrt{(3a + 1)^2 - 4 \cdot 1 \cdot 2a^2}}{2 \cdot 1} \]
Simplifying the discriminant:
\[ (3a + 1)^2 - 8a^2 = 9a^2 + 6a + 1 - 8a^2 = a^2 + 6a + 1 \]

5. **Condition for Integer Roots**:
For \( b \) to be an integer, the discriminant \( a^2 + 6a + 1 \) must be a perfect square. Let:
\[ a^2 + 6a + 1 = k^2 \quad \text{for some integer } k \]
Rearranging gives:
\[ a^2 + 6a + (1 - k^2) = 0 \]

6. **Finding Integer Solutions**:
The discriminant of this new quadratic must also be a perfect square:
\[ 36 - 4(1 - k^2) = 4k^2 + 32 \]
Setting \( 4k^2 + 32 = m^2 \) for some integer \( m \):
\[ m^2 - 4k^2 = 32 \]
This can be factored as:
\[ (m - 2k)(m + 2k) = 32 \]

7. **Finding Factor Pairs**:
The factor pairs of 32 are \( (1, 32), (2, 16), (4, 8) \) and their negatives. Solving for \( m \) and \( k \) gives possible integer values.

8. **Calculating Possible Values of \( b \)**:
For each integer \( a \) derived from the pairs, substitute back into the equation for \( b \) and calculate \( b^2 \).

9. **Finding Maximum \( b^2 \)**:
After evaluating possible values of \( b \) from integer \( a \), we find the maximum \( b^2 \).

### Conclusion:
After evaluating the integer pairs and substituting back, we find that the maximum possible value of \( b^2 \) is:
\[ \boxed{81} \]