Example 4 Solve the equation
$$
\sqrt{x^{2}+x+1}-\sqrt{x^{2}}+7 x+5=3 x+2
$$
Given: $\because \sqrt{x^{2}+x} \div 1, \frac{3 x+2}{2}$, $-\sqrt{x^{2}+7 x+5}$ form an arithmetic sequence, we set
$$
\left\{\begin{array}{l}
\sqrt{x^{2}+x+1}=\frac{3 x+2}{2}-d, \\
\sqrt{x^{2}+7 x+5}=-\frac{3 x+2}{2}-d .
\end{array}\right.
$$
$(1)^{2}-(2)^{2}$, simplifying gives
$$
\begin{array}{l}
-2(3 x+2)=-2(3 x+2) d \\
\Rightarrow(3 x+2)(1-d)=0 \\
\Rightarrow x=-\frac{2}{3}, d=1 .
\end{array}
$$

Substituting $d=1$ into (1), we get
$$
\sqrt{x^{2}+x+1}=\frac{3}{2} x \Rightarrow 5 x^{2}-4 x-4=0 \text {. }
$$

Solving gives $x=\frac{2 \pm 2 \sqrt{6}}{5}$ (extraneous roots).
Upon verification, the solution to the original equation is $x=-\frac{2}{3}$.
In general, for irrational equations of the form $\sqrt{f(x)} \pm \sqrt{g(x)}=$ $F(x)$, and $f(x)-g(x)=k \cdot F(x)$ (where $k$ is a non-zero constant), the method of setting the common difference can be used to solve them.