Since the sum of two squares is always non-negative, the only way for the sum of the two squares to be equal to zero is if each individual square is equal to zero. Therefore:

x^2 + 2xy + y^2 = 0
and
x^2 - 5y - 1 = 0

The first equation can be factored as (x+y)^2 = 0, which means x+y = 0. Substituting this into the second equation:

x^2 - 5(x+y) - 1 = 0
x^2 - 5(0) - 1 = 0
x^2 = 1
x = ±1

Now substitute the values of x into x+y = 0:
1+y = 0
y = -1
or
-1+y = 0
y = 1

Therefore, the solutions to the given equation are (x,y) = (1,-1) and (x,y) = (-1,1).