To solve the given equation, we need to isolate the square roots on one side and the x term on the other side. Let's start by moving the term involving the square roots to one side of the equation:

[tex] \sqrt{3x + 4} - \sqrt{x - 4} = 2 \sqrt{x} [/tex]

Add [tex] \sqrt{x - 4} [/tex] to both sides:

[tex] \sqrt{3x + 4} = 2 \sqrt{x} + \sqrt{x - 4} [/tex]

Now, square both sides of the equation to eliminate the square roots:

[tex] (3x + 4) = (2\sqrt{x} + \sqrt{x - 4})^2 [/tex]

Expand the right side of the equation:

[tex] 3x + 4 = 4x + 4 + 4 \sqrt{x(x-4)} + x - 4 [/tex]
[tex] 3x + 4 = 5x + 4 + 4\sqrt{x(x-4)} [/tex]
[tex] -2x = 4\sqrt{x(x-4)} [/tex]
[tex] -2x^2 = 16(x-4) [/tex]
[tex] -2x^2 = 16x - 64 [/tex]
[tex] 2x^2 + 16x - 64 = 0 [/tex]

Solving the equation [tex] 2x^2 + 16x - 64 = 0 [/tex] gives two possible solutions for x, which are:

x = 4 and x = -16

However, since the original equation contains square roots, we need to check whether these solutions satisfy the domain of the function. By plugging in x = 4, we see that it does not lead to any subtraction under a square root sign, which is not allowed. Therefore, the only valid solution for the given equation is x = -16.