[Example 2.2.1] Simplify: $M=\frac{2}{\sqrt{4-3 \sqrt[4]{5}+2 \sqrt[4]{25}-\sqrt[4]{125}}}$.
This is a problem of simplifying an irrational expression. Such problems are usually cumbersome to compute, but observing the expression to be simplified, there are $\sqrt[4]{25}, \sqrt[4]{125}$, which have the relationship
$$
\sqrt[4]{25}=(\sqrt[4]{5})^{2}, \sqrt[4]{125}=(\sqrt[4]{5})^{3} .
$$

This inspires us to consider using a more abstract, more general letter $x$ to substitute for $\sqrt[4]{5}$, making the expression to be simplified a rational expression, thus turning it into a problem of simplifying fractions.
For this purpose, let $\sqrt[4]{5}=x$, then
$$
\sqrt[4]{25}=x^{2}, \sqrt[4]{125}=x^{3} .
$$

When simplifying the fraction below, due to the special nature of $x$, when encountering $x^{4}$, replace it with 5, when encountering $x^{5}$, replace it with $5 x$, and when encountering $x^{6}$, replace it with $5 x^{2}$.
Thus, the original expression becomes
$$
\begin{aligned}
M^{2} & =\frac{4}{4-3 x+2 x^{2}-x^{3}} \\
& =-\frac{4}{x^{3}-2 x^{2}+3 x-4} \\
& =-\frac{4}{\left(x^{3}+3 x\right)-\left(2 x^{2}+4\right)} \\
& =-\frac{4\left[\left(x^{3}+3 x\right)+\left(2 x^{2}+4\right)\right]}{\left[\left(x^{2}+3 x\right)-\left(2 x^{2}+4\right)\right]\left[\left(x^{2}+3 x\right)+\left(2 x^{2}+4\right)\right]} \\
& =-\frac{4\left(x^{3}+2 x^{2}+3 x+4\right)}{\left(x^{3}+3 x\right)^{2}-\left(2 x^{2}+4\right)^{2}} \\
& =-\frac{4\left(x^{3}+2 x^{2}+3 x+4\right)}{x^{6}+6 x^{4}+9 x^{2}-4 x^{4}-16 x^{2}-16} \\
& =-\frac{4\left(x^{3}+2 x^{2}+3 x+4\right)}{5 x^{2}+30+9 x^{2}-20-16 x^{2}-16} \\
& =-\frac{4\left(x^{3}+2 x^{2}+3 x+4\right)}{-2 x^{2}-6} \\
& =\frac{2\left(x^{3}+2 x^{2}+3 x+4\right)}{x^{2}+3} \\
& =\frac{2 x\left(x^{2}+3\right)+4\left(x^{2}+2\right)}{x^{2}+3} \\
& =2 x+4-\frac{4}{x^{2}+3} \\
& =2 x+4+x^{2}-3 \\
& =(x+1)^{2} . \\
& =2 x+4-\frac{4\left(x^{2}-3\right)}{\left(x^{2}+3\right)\left(x^{2}-3\right)} \\
&
\end{aligned}
$$

Since $M>0$, we have
$$
M=x+1=\sqrt[1]{5}+1 .
$$

This problem transforms a specific numerical problem into a problem of letter operations through generalization. In the process of generalization, it also takes into account the special nature of the letter, which is a specific number, transforming a special but difficult-to-calculate problem into a general but easy-to-calculate problem, which is the key to solving this problem.