6.277 \sqrt{x-1}+\sqrt{x+3}+2 \sqrt{(x-1)(x+3)}=4-2 x
$$

6.277 \sqrt{x-1}+\sqrt{x+3}+2 \sqrt{(x-1)(x+3)}=4-2 x
$$
Solution. Let $u=\sqrt{x-1}, v=\sqrt{x+3}$, then $u+v+2uv=-(u^2+v^2)+6 \Leftrightarrow$ $(u+v)^2+(u+v)-6=0 \Leftrightarrow\left[\begin{array}{l}u+v=-3, \\ u+v=2\end{array} \Leftrightarrow\left[\begin{array}{l}\varnothing, \\ \sqrt{x-1}+\sqrt{x+3}=2 .\end{array}\right.\right.$

Since $x \geq 1$, then $\sqrt{x+3} \geq 2 \Rightarrow x=1$ is the only solution to the last equation.

Answer: $x=1$.