$6.278 \sqrt{2 x+3}+\sqrt{x+1}=3 x+2 \sqrt{2 x^{2}+5 x+3}-16$.

Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.

$6.278 \sqrt{2 x+3}+\sqrt{x+1}=3 x+2 \sqrt{2 x^{2}+5 x+3}-16$.
Solution. $\sqrt{2 x+3}+\sqrt{x+1}=3 x-16+2 \sqrt{(2 x+3)(x+1)}$.

Let $u=\sqrt{2 x+3}, v=\sqrt{x+1}$, then $u+\dot{v}=u^{2}+v^{2}-20+2 u v \Leftrightarrow$

$$
\Leftrightarrow(u+v)^{2}-(u+v)-20=0 . \Leftrightarrow\left[\begin{array}{l}
u+v=5, \\
u+v=-4
\end{array} \Leftrightarrow \sqrt{2 x+3}+\sqrt{x+1}=5 \Leftrightarrow\right.
$$

$$
\begin{aligned}
& \Leftrightarrow\left\{\begin{array} { l } 
{ x \geq - 1 , } \\
{ 3 x + 4 + 2 \sqrt { ( 2 x + 3 ) ( x + 1 ) } = 2 5 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
x \geq-1, \\
x \leq 7, \\
4(2 x+3)(x+1)=9(7-x)^{2}
\end{array} \Leftrightarrow\right.\right. \\
& \Leftrightarrow\left\{\begin{array}{l}
-1 \leq x \leq 7, \\
x^{2}-146 x+429=0
\end{array} \Leftrightarrow x=3 .\right.
\end{aligned}
$$

Answer: $x=3$.