Example 11. Solve the equation

$$
\sqrt{x^{2}+3 x-4}=\sqrt{2 x+2}
$$
Solution. Solving the system of inequalities

$$
\left\{\begin{array}{l}
x^{2}+3 x-4 \geqslant 0 \\
2 x+2 \geqslant 0
\end{array}\right.
$$

we find the domain of the equation: $x \geqslant 1$.

By squaring both sides of the original equation, we obtain the equation

$$
x^{2}+3 x-4=2 x+2
$$

which is a consequence of it. The roots of this equation are: $x_{1}=2$ and $x_{2}=-3$.

The root $x_{2}=-3$ does not belong to the domain of the original equation and is therefore extraneous.

Substituting the number 2 into the original equation, we get a true equality; hence, $x=2$ is its only root.