(x+1)(y+1) = 2xy
Multiply the left side out:
xy + x + y + 1 = 2xy
Subtract xy from both sides:
x + y + 1 = xy
Isolate the terms in y
x + 1 = xy - y
Factor y out on the right
x + 1 = y(x - 1)
Divide both sides by (x-1)
 = y
I like y on the left
y = 
Divide that fraction out by long division

      1
x-1)x+1
    x-1
      2

y = QUOTIENT + 
y = 1 + 
In order for y to be an integer, the fractional expression
 must equal to an integer.  That means the 
denominator x-1 must be a factor of the numerator, 2.

The only factors of 2 are 1 and 2. So we set the
denominator x-1 equal to each of those:

If x-1 = 1
     x = 2

y = 
y =  
y = 
y = 3

So one possible pair of integers is x=2 and y=3.

If x-1 = 2
     x = 3

y = 
y =  
y = 
y = 2

So the other possible pair of integers is x=3 and y=2.

Edwin