Solution:

To find the values of x and y that satisfy the given equation x^2 + 84x + 2008 = y^2, we can use a technique called completing the square.

Completing the square:
1. Rewrite the equation by moving the constant term to the right side: x^2 + 84x = y^2 - 2008.
2. Take half of the coefficient of x (which is 42) and square it: (42)^2 = 1764.
3. Add the result to both sides of the equation: x^2 + 84x + 1764 = y^2 - 2008 + 1764.
4. Simplify both sides: (x + 42)^2 = y^2 - 244.
5. Rearrange the equation to isolate the perfect square term: (x + 42)^2 - y^2 = -244.
6. Use the difference of squares formula to factorize the left side: (x + 42 + y)(x + 42 - y) = -244.

Factoring the right side:
1. Find the prime factorization of -244: -244 = -1 * 2^2 * 61.
2. Since the factors have different signs, we can rewrite the equation as follows: (x + 42 + y)(x + 42 - y) = -1 * 2^2 * 61.

Case 1: (x + 42 + y) = -1 and (x + 42 - y) = 2^2 * 61
1. Solve the first equation for y: y = -x - 43.
2. Substitute this value of y into the second equation: (x + 42 - (-x - 43)) = 4 * 61.
3. Simplify the equation: 2x + 85 = 244.
4. Solve for x: 2x = 159, x = 79.5 (which is not a positive integer).

Case 2: (x + 42 + y) = 2 and (x + 42 - y) = -1 * 2 * 61
1. Solve the first equation for y: y = 2 - x - 42, y = -x - 40.
2. Substitute this value of y into the second equation: (x + 42 - (-x - 40)) = -2 * 61.
3. Simplify the equation: 2x + 2 = -122.
4. Solve for x: 2x = -124, x = -62 (which is not a positive integer).

Case 3: (x + 42 + y) = -2 and (x + 42 - y) = 2 * 61
1. Solve the first equation for y: y = -2 - x - 42, y = -x - 44.
2. Substitute this value of y into the second equation: (x + 42 - (-x - 44)) = 2 * 61.
3. Simplify the equation: 2x + 86 = 122.
4. Solve for x: 2x =