There are no integer solutions.
Since a^2, b^2, c^2, and d^2 are all positive, cd has to be greater than or equal to c^2+d^2. If c is larger than d, then c^2 is larger than cd and with d^2 it's even larger. If d is larger than c, then d^2 is larger than cd and with c^2 it's even larger. Therefore they must be equal. But if they are equal cd=c^2 so d^2=0 meaning d=0, c=0, and cd=0. So a^2+b^2+0+0=0. Now, since a^2 and b^2 are both positive and they add to 0, they both must equal 0. The only problem is 2500 (a+50)(b+50) is positive when a and b are both 0, therefore there are no integer solutions.
I am sorry for confusion, this is a system of equations [a^2 + b^2 + c^2 + d^2 = 2500] and the second equation is [(a+50)(b+50) = cd].