Solution:

Given, x² + bx + 72 = 0

As the equation has two distinct integer roots, the discriminant b² – 4ac must be a perfect square.

b² – 4ac = b² – 4(1)(72) = b² – 288

Since the discriminant is a perfect square, we can write:

b² – 288 = k²

where k is a positive integer.

Rearranging the above equation, we get:

b² – k² = 288

Using the difference of squares factorization, we can write:

(b + k)(b – k) = 288

Since b + k and b – k have the same parity, they are both even or both odd.

Case 1: b + k and b – k are both even

Let b + k = 2m and b – k = 2n, where m and n are integers.

Substituting in the above equation, we get:

4mn = 288

The pairs of factors of 288 are:

1, 288; 2, 144; 3, 96; 4, 72; 6, 48; 8, 36; 9, 32; 12, 24

For each pair of factors, we get two equations:

m + n = factor and m – n = factor

Solving for m and n, we get:

m = (sum of factors)/4 and n = (difference of factors)/4

Since b = m + n, we can find all the possible values of b.

For example, when the factors are 4 and 72, we get m = 19 and n = -15, so b = 4.

Therefore, in this case, there are 12 possible values of b.

Case 2: b + k and b – k are both odd

Let b + k = 2m + 1 and b – k = 2n + 1, where m and n are integers.

Substituting in the above equation, we get:

4mn + 2m + 2n = 288

Dividing by 2, we get:

2mn + m + n = 144

Using Simon's Favorite Factoring Trick, we can write:

(2m + 1)(2n + 1) = 289

Since 289 is a perfect square, we can write:

(2m + 1)(2n + 1) = 17²

The factors of 17² are:

1, 289

17, 17

Therefore, we get two equations:

2m + 1 = 17 and 2n + 1 = 1

or

2m + 1 = 1 and 2n + 1 = 17

Solving for m and n, we get:

m = 8 and n = -9

or

m = -9 and n = 8

Since b = m + n, we can find all the possible values of b.

For example, when m = 8 and n = -9, we get b = -1.

Therefore, in this case, there are 2 possible values of b.

Total number of possible values of b = 12 + 2 = 14.

However,