$2.55 \sqrt{x+1}+\sqrt{4 x+13}=\sqrt{3 x+12}$.

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

$2.55 \sqrt{x+1}+\sqrt{4 x+13}=\sqrt{3 x+12}$.
2.55 By squaring both sides of the equation, we get

$$
\begin{aligned}
& x+1+4 x+13+2 \sqrt{(x+1)(4 x+13)}=3 x+12 \\
& \sqrt{(x+1)(4 x+13)}=-(x+1)
\end{aligned}
$$

Another squaring would eliminate the irrationality, but there is no need for this transformation here. We notice that the derived equation can have a solution only under the condition $x+1 \leqslant 0$. At the same time, one of the conditions for the existence of a solution to the original equation is the requirement $x+1 \geqslant 0$. Both these conditions are consistent in the unique case where $x+1=0$, from which $x=-1$. This value of $x$, as is easily verified, satisfies the original equation. Since the derived equation has no other roots, the original equation also has no other roots. Thus, $x=-1$.

Answer: $x=-1$.