Example 4 (2004 National College Entrance Examination for Liberal Arts) Given the function $f(x)=a x^{3}+3 x^{2}-x+1$ is decreasing on $\mathbf{R}$, find the range of values for $a$.
Solve for the derivative of the function $f(x)$: $f^{\prime}(x)=3 a x^{2}+6 x-1$.
(1) When $f^{\prime}(x)-3$, there exists an interval on $\mathrm{R}$ where $f^{\prime}(x)>0$.
Therefore, when $a>-3$, the function $f(x)(x \in \mathrm{R})$ is not a decreasing function.
In conclusion, the range of values for $a$ is $(-\infty,-3]$.