Example 7 The equation in terms of $x$
$$
x^{3}-a x^{2}-2 a x+a^{2}-1=0
$$

has only one real root. Then the range of values for $a$ is
Solve for $a$ as the main variable, to get the equation in terms of $a$
$$
\begin{array}{l}
a^{2}-\left(x^{2}+2 x\right) a+x^{3}-1=0 \\
\Rightarrow[a-(x-1)]\left[a-\left(x^{2}+x+1\right)\right]=0 \\
\Rightarrow x=a+1 \text { or } x^{2}+x+(1-a)=0 .
\end{array}
$$

Since the original equation has only one real root, we have
$$
\Delta=1-4(1-a)<0 \Rightarrow a<\frac{3}{4} \text {. }
$$