To solve the system of equations:

1) Start with the first equation: U - 2X = 3√5.
We can rewrite it as U = 2X + 3√5

2) Substitute U in the second equation: X^2 - X(2X + 3√5) - (2X)^2 = -5
Simplify the expression:
X^2 - 2X^2 - 3X√5 - 4X^2 = -5
-5X^2 - 3X√5 = -5
-X^2 - 3X√5 = -1
X^2 + 3X√5 = 1

3) We now have the system of equations:
U = 2X + 3√5
X^2 + 3X√5 = 1

This system of equations is nonlinear and may require a more advanced approach to solve. Let's attempt to solve it implicitly by eliminating the variable U:

If U = 2X + 3√5, then substitute this into the second equation: (2X + 3√5)^2 + 3(2X + 3√5)√5 = 1

Expand and simplify the equation to solve for X.