Example 2 Solve the functional inequality $f(-x) \leqslant x^{4}-2 x^{2}+3 x-1$.


Given the function $p(x)(\forall p(x) \leqslant 0, x \in \mathbf{R})$, let $f(-x)=x^{1}-2 x^{2}+3 x-1+p(x)$. Then
$$
f(-x)=(-x)^{4}-2(-x)^{2}-3(-x)-1+p(-(-x))
$$

So $f(x)=x^{4}-2 x^{2}-3 x-1+p(-x)$, where $\forall p(x) \leqslant 0, x \in \mathbf{R}$.
Analysis: Since the introduction of the auxiliary function is an equivalent transformation, it is not necessary to substitute it back into the original inequality for verification. However, it is still essential to verify the solution of the function equation obtained after introducing the auxiliary function. After introducing the auxiliary function, the solution of the obtained function equation sometimes requires appropriate substitution of variables (the substitution process should ensure that the domain of the function does not change), resulting in one or several new function equations. Then, these new equations are combined with the original equation, and the solution of the original function equation is obtained through elimination.
The conclusion of this problem can also be succinctly written as: $f(x) \leqslant x^{4}-2 x^{2}-3 x-1, x \in \mathbf{R}$.