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It is not clear in your question on which field or ring you want to solve your equation. Though the following sequence of computations seems to indicate that your system is verified by any value for x and y:

sage: var('x,y')
(x, y)
sage: p = -6-4*y-x-(1+y)*x+sqrt((4*(1+y))*(2+x)*(4+2*y+x)+(-(1+y)*x+2+x)^2)
sage: q = (2*(4+2*y+x))*(1+y)-(1+y)*x+2+x+sqrt((4*(1+y))*(2+x)*(4+2*y+x)+(-(1+y)*x+2+x)^2)-(2+y)*(-(1+y)*x+2+x+sqrt((4*(1+y))*(2+x)*(4+2*y+x)+(-(1+y)*x+2+x)^2))
sage: s = solve(p==0,x,y); s
([x == -(4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)/(y + 2)],
 [1])
sage: x0 = s[0][0].rhs()
sage: q.subs(x=x0)
-((4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)*(y + 1)/(y + 2) + sqrt(-4*(2*y - (4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)/(y + 2) + 4)*(y + 1)*((4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)/(y + 2) - 2) + ((4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)*(y + 1)/(y + 2) - (4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)/(y + 2) + 2)^2) - (4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)/(y + 2) + 2)*(y + 2) + 2*(2*y - (4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)/(y + 2) + 4)*(y + 1) + (4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)*(y + 1)/(y + 2) + sqrt(-4*(2*y - (4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)/(y + 2) + 4)*(y + 1)*((4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)/(y + 2) - 2) + ((4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)*(y + 1)/(y + 2) - (4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)/(y + 2) + 2)^2) - (4*y - sqrt((x^2 + 8*x + 16)*y^2 + 4*x^2 + 4*(x^2 + 7*x + 12)*y + 24*x + 36) + 6)/(y + 2) + 2
sage: q.subs(x=x0).simplify_rational() # or .simplify_full()
0

I did not try to understand why this holds.