x^2 - y^2 = 2026 -2(x+2y)

=> x^2 +2x -(y^2 -4y) = 2026

=> (x+1)^2-1 -{(y-2)^2-4} = 2026

= > (x+1)^2 - (y-2)^2 = 2023

=> (x+y-1)(x-y+2) = 289*7 = 119*17 = 2023*1

=> (x+y-1, x-y+3) = (289,7), (119, 17), (2023, 1)

=> (x+y, x-y) = (290, 4), (120, 14), (2024, --2)

=> (x,y) = (147, 143), (67, 58), (1011, 1013)

However, we observe that 67, 1013 are prime numbers.

Therefore, (x, y) = (147, 143) is the only possible solution.

Consequently, x*y = 147*143 = 21021.

*Edited on ***February 29, 2024, 8:55 am**