\(x=\frac{\sqrt{\left(2-\sqrt{3}\right)^2}}{2}=\frac{2-\sqrt{3}}{2}=\frac{4-2\sqrt{3}}{4}=\left(\frac{\sqrt{3}-1}{2}\right)^2\)
\(\Rightarrow\sqrt{x}=\frac{\sqrt{3}-1}{2}\)
\(\frac{4\sqrt{x}}{\left(\sqrt{x}+1\right)^2}=\frac{2\left(\sqrt{3}-1\right)}{\left(\frac{\sqrt{3}-1}{2}+1\right)^2}=\frac{8\left(\sqrt{3}-1\right)}{\left(\sqrt{3}+1\right)^2}=-20+12\sqrt{3}\)