Example 18. Solve the equation

$$
x=(\sqrt{1+x}+1)(\sqrt{10+x}-4)
$$
Solution. The domain of definition (DOD) of equation (6): $x \geqslant-1$. Multiplying both sides of equation (6) by the expression $\sqrt{1+x}-1$, we obtain the equation

$$
x(\sqrt{1+x}-\sqrt{10+x}+3)=0
$$

which is a consequence of equation (6). This equation has two roots: $x_{1}=0$ and $x_{2}=-1$ (note that the DOD has not been extended). Both found roots belong to the DOD of equation (6); nevertheless, checking shows that $x_{1}=0$ is an extraneous root of equation (6), while $x_{2}=-1$ satisfies it.

Thus, equation (6) has a unique root $x=-1$.