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 analyze the equation \((x^2 + ax + 20)(x^2 + 17x + b) = 0\) and ensure that all roots are negative integers.

### Step 1: Analyze the first quadratic \(x^2 + ax + 20 = 0\)

Let the roots of this quadratic be \(p\) and \(q\). By Vieta's formulas, we have:
- \(p + q = -a\)
- \(pq = 20\)

Since \(p\) and \(q\) are negative integers, we need to find pairs of negative integers whose product is 20. The pairs \((p, q)\) that satisfy this are:
- \((-1, -20)\)
- \((-2, -10)\)
- \((-4, -5)\)

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

### Step 2: Analyze the second quadratic \(x^2 + 17x + b = 0\)

Let the roots of this quadratic be \(r\) and \(s\). Again, using Vieta's formulas:
- \(r + s = -17\)
- \(rs = b\)

Since \(r\) and \(s\) are also negative integers, we need to find pairs of negative integers that sum to -17. The pairs \((r, s)\) that satisfy this 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 \((-1, -16)\):
- \(rs = (-1)(-16) = 16 \Rightarrow b = 16\)
2. For \((-2, -15)\):
- \(rs = (-2)(-15) = 30 \Rightarrow b = 30\)
3. For \((-3, -14)\):
- \(rs = (-3)(-14) = 42 \Rightarrow b = 42\)
4. For \((-4, -13)\):
- \(rs = (-4)(-13) = 52 \Rightarrow b = 52\)
5. For \((-5, -12)\):
- \(rs = (-5)(-12) = 60 \Rightarrow b = 60\)
6. For \((-6, -11)\):
- \(rs = (-6)(-11) = 66 \Rightarrow b = 66\)
7. For \((-7, -10)\):
- \(rs = (-7)(-10) = 70 \Rightarrow b = 70\)
8. For \((-8, -9)\):
- \(rs = (-8)(-9) = 72 \Rightarrow b = 72\)

### Step 3: Calculate \(a + b\)

Now we will calculate \(a + b\) for each combination of \(a\) and \(b\):

1. For \(a = 21\):
- \(a + b = 21 + 16 = 37\)
- \(a + b = 21 + 30 = 51\)
- \(a + b = 21 + 42 = 63\)
- \(a + b = 21 + 52 = 73\)
- \(a + b = 21 + 60 = 81\)
- \(a + b = 21 + 66 = 87\)
- \(a + b = 21 + 70 = 91\)
- \(a + b = 21 + 72 = 93\)

2. For \(a = 12\):
- \(a + b = 12 + 16 = 28\)
- \(a + b = 12 + 30 = 42\)
- \(a + b = 12 + 42 = 54\)
- \(a + b = 12 + 52 = 64\)
- \(a + b = 12 + 60 = 72\)
- \(a + b = 12 + 66 = 78\)
- \(a + b = 12 + 70 = 82\)
- \(a + b = 12 + 72 = 84\)

3. For \(a = 9\):
- \(a + b = 9 + 16 = 25\)
- \(a + b = 9 + 30 = 39\)
- \(a + b = 9 + 42 = 51\)
- \(a + b = 9 + 52 = 61\)
- \(a + b = 9 + 60 = 69\)
- \(a + b = 9 + 66 = 75\)
- \(a + b = 9 + 70 = 79\)
- \(a + b = 9 + 72 = 81\)

### Step 4: Find the minimum value of \(a + b\)

From the calculations, the smallest value of \(a + b\) is \(25\) when \(a = 9\) and \(b = 16\).

### Final Answer
The smallest possible value of \(a + b\) is **25**.