To solve this equation, we can start by simplifying each square root separately:

Simplify √(3x+9-4√(3x+5)):
Let y = √(3x+5)
Therefore, y^2 = 3x + 5
Now, the given equation becomes:
√(y^2 + 4) = 1
y^2 + 4 = 1
y^2 = -3, which is not possible since the square root of a number cannot be negative. Therefore, there are no solutions for this part of the equation.

Next, we simplify √(3x+14-6√(3x+5)):
Let z = √(3x+5)
Therefore, z^2 = 3x + 5
Now, the given equation becomes:
√(2z+9) = 1
2z + 9 = 1
2z = -8
z = -4
Since the square root of a number cannot be negative, there are no solutions for this part of the equation either.

Therefore, the original equation has no solutions.