To solve the equation, we start by simplifying each term:
- The first term is \( 2 \tan^{-1} x \). We will consider the inverse tangent function and express \( \tan^{-1} x \) in terms of an angle \( \theta \), where \( \tan \theta = x \).
- The second term involves \( \sin^{-1} \left( \frac{2x}{1 + x^2} \right) \), which is a standard identity for \( \sin^{-1} \left( \frac{2x}{1 + x^2} \right) \), which equals \( \tan^{-1} x \).
Thus, the equation becomes:
\[
2 \tan^{-1} x + \tan^{-1} x = 4\sqrt{3}
\]
Simplifying further:
\[
3 \tan^{-1} x = 4\sqrt{3}
\]
\[
\tan^{-1} x = \frac{4\sqrt{3}}{3}
\]
Now, take the tangent of both sides:
\[
x = \tan \left( \frac{4\sqrt{3}}{3} \right)
\]
The solution gives \( x = 1 \).
