(√x + 3 )+ √(3x + 1) = 2√x + căn(2 x+2)

To solve the equation (√x + 3) + √(3x + 1) = 2√x + √(2x + 2), we can follow these steps:

1. Start by isolating the square root terms on one side of the equation. Subtract 2√x from both sides:
(√x + 3) + √(3x + 1) - 2√x = √(2x + 2)

2. Simplify the left side of the equation by combining like terms:
√x - 2√x + 3 + √(3x + 1) = √(2x + 2)

3. Combine the √x and -2√x terms:
-√x + 3 + √(3x + 1) = √(2x + 2)

4. Move the √(3x + 1) term to the right side of the equation by subtracting it from both sides:
-√x + 3 = √(2x + 2) - √(3x + 1)

5. To eliminate the square roots on the right side, we can square both sides of the equation:
(-√x + 3)^2 = (√(2x + 2) - √(3x + 1))^2

6. Expand both sides of the equation:
(√x)^2 - 2(-√x)(3) + (3)^2 = (2x + 2) - 2√(2x + 2)√(3x + 1) + (3x + 1)

7. Simplify both sides of the equation:
x - 6√x + 9 = 2x + 2 - 2√(6x^2 + 8x + 2) + 3x + 1

8. Combine like terms on both sides of the equation:
x - 6√x + 9 = 5x + 3 - 2√(6x^2 + 8x + 2)

9. Move all terms involving √x to the left side of the equation and all other terms to the right side:
-5x + 2√(6x^2 + 8x + 2) = -6√x + 5x - 6

10. Combine like terms on both sides of the equation:
2√(6x^2 + 8x + 2) = -6√x - 6

11. Divide both sides of the equation by 2:
√(6x^2 + 8x + 2) = -3√x - 3

12. Square both sides of the equation to eliminate the square root:
(√(6x^2 + 8x + 2))^2 = (-3√x - 3)^2

13. Expand both sides of the equation:
6x^2 + 8x + 2 = 9x^2 + 18√x + 9

14. Move all terms to one side of the equation:
9x^2 - 6x^2 - 8x - 18√x + 9 - 2 = 0

15. Combine like terms:
3x^2 - 8x - 18√x + 7 = 0

This is the final equation after simplifying. It cannot be solved algebraically, so numerical methods or approximations would be needed to find the solutions.