To solve this equation, we can start by simplifying both sides of the equation.

On the left side, we have 2√3x - 4√12x. We can simplify √12x by breaking it down into its prime factors. The prime factorization of 12 is 2 * 2 * 3. Therefore, we can rewrite √12x as √(2 * 2 * 3 * x) = 2√3√x.

So, the left side of the equation becomes 2√3x - 4(2√3√x) = 2√3x - 8√3√x.

On the right side, we have 12 - √75x.

Now, we can rewrite the equation as:

2√3x - 8√3√x = 12 - √75x.

Next, we can combine like terms on both sides of the equation. On the left side, we have 2√3x - 8√3√x. Since both terms have a common factor of √3x, we can factor it out:

√3x(2 - 8√3) = 12 - √75x.

Now, we can simplify further by multiplying out the common factor on the left side:

2 - 8√3 = 12 - √75x.

Next, we can isolate the term with the variable on one side of the equation. Let's move the term with the variable, -√75x, to the left side by adding √75x to both sides:

2 - 8√3 + √75x = 12.

Now, we can simplify further by combining like terms on the left side:

2 - 8√3 + √75x = 12.

Finally, we can solve for x by isolating the variable term. Let's move the constant term, 2 - 8√3, to the right side by subtracting it from both sides:

√75x = 12 - 2 + 8√3.

Simplifying further:

√75x = 10 + 8√3.

To solve for x, we can square both sides of the equation:

(√75x)^2 = (10 + 8√3)^2.

75x = 100 + 160√3 + 64(3).

Simplifying further:

75x = 100 + 160√3 + 192.

75x = 292 + 160√3.

Finally, we can solve for x by dividing both sides of the equation by 75:

x = (292 + 160√3)/75.

So, the solution to the equation is x = (292 + 160√3)/75.