Let a, b be integers such that all the roots of the equation (`x^2 + ax + 20)(x^2 + 17x + b) = 0` are negative integers. What is the smallest possible value of a + b ?

Answer

To solve the problem, we need to find integers \( a \) and \( b \) such that all roots of the equation

\[ (x^2 + ax + 20)(x^2 + 17x + b) = 0 \]

are negative integers.

### Step 1: Analyze the first quadratic equation

The first factor is \( x^2 + ax + 20 \). Let the roots of this equation be \( -\alpha \) and \( -\beta \), where \( \alpha \) and \( \beta \) are positive integers.

Using Vieta's formulas:
- The sum of the roots \( -\alpha - \beta = -a \) implies \( \alpha + \beta = a \).
- The product of the roots \( (-\alpha)(-\beta) = \alpha \beta = 20 \).

### Step 2: Find pairs of factors of 20

The pairs of positive integers \( (\alpha, \beta) \) that multiply to 20 are:
- \( (1, 20) \)
- \( (2, 10) \)
- \( (4, 5) \)

Now, we calculate \( a \) for each pair:
1. For \( (1, 20) \): \( a = 1 + 20 = 21 \)
2. For \( (2, 10) \): \( a = 2 + 10 = 12 \)
3. For \( (4, 5) \): \( a = 4 + 5 = 9 \)

### Step 3: Analyze the second quadratic equation

The second factor is \( x^2 + 17x + b \). Let the roots of this equation be \( -p \) and \( -q \), where \( p \) and \( q \) are positive integers.

Using Vieta's formulas:
- The sum of the roots \( -p - q = -17 \) implies \( p + q = 17 \).
- The product of the roots \( pq = b \).

### Step 4: Find pairs of factors that add up to 17

The pairs of positive integers \( (p, q) \) that add up to 17 are:
- \( (1, 16) \)
- \( (2, 15) \)
- \( (3, 14) \)
- \( (4, 13) \)
- \( (5, 12) \)
- \( (6, 11) \)
- \( (7, 10) \)
- \( (8, 9) \)

Now, we calculate \( b \) for each pair:
1. For \( (1, 16) \): \( b = 1 \times 16 = 16 \)
2. For \( (2, 15) \): \( b = 2 \times 15 = 30 \)
3. For \( (3, 14) \): \( b = 3 \times 14 = 42 \)
4. For \( (4, 13) \): \( b = 4 \times 13 = 52 \)
5. For \( (5, 12) \): \( b = 5 \times 12 = 60 \)
6. For \( (6, 11) \): \( b = 6 \times 11 = 66 \)
7. For \( (7, 10) \): \( b = 7 \times 10 = 70 \)
8. For \( (8, 9) \): \( b = 8 \times 9 = 72 \)

### Step 5: Find the smallest possible value of \( a + b \)

Now we have:
- Possible values of \( a \): \( 21, 12, 9 \)
- Possible values of \( b \): \( 16, 30, 42, 52, 60, 66, 70, 72 \)

To find the smallest possible value of \( a + b \):
1. If \( a = 21 \), then \( a + b \) can be \( 21 + 16 = 37 \).
2. If \( a = 12 \), then \( a + b \) can be \( 12 + 16 = 28 \).
3. If \( a = 9 \), then \( a + b \) can be \( 9 + 16 = 25 \).

The smallest value of \( a + b \) is \( 25 \).

### Final Answer

Thus, the smallest possible value of \( a + b \) is

\[ \boxed{25} \]