9
b + 4)9b + 63
      9b + 36
           27



b+4 can equal any positive or negative divisor of 27.

The divisors of 27 are

1, -1, 3, -3, 9, -9, -27, 27

There are 8 of them, so the answer is 8.

That's because each one when substituted for b+4 produces 
a pair of integers that satisfies 

For example, say, when b+4 = -9, b = -13 and substituting
b = -13 into



gives







, so the pair (b,c) = (-13,6) is a pair of integers 

that satisfies .

So the answer is 8.

Edwin