Suppose the derivative of a function $f$ is
\[
f^{\prime}(x)=(x+2)(x-4)^{2}(x-1)^{3} .
\]
Then $f(x)$ is decreasing on the following interval:
$(1, \infty)$
$(-\infty, 1)$
$(1,4)$
$(-\infty,-2)$
$(-2,1)$
Thus, the function $f(x)$ is decreasing on the interval $\boxed{(-2,1)}$.
Step 1 :Given the derivative of a function $f$ is $f^{\prime}(x)=(x+2)(x-4)^{2}(x-1)^{3}$.
Step 2 :The function $f(x)$ is decreasing where its derivative $f'(x)$ is negative. The derivative is a product of several factors, and it will be negative where an odd number of these factors are negative.
Step 3 :We can find the intervals where each factor is negative, and then combine these intervals to find where the product is negative.
Step 4 :The roots of the derivative are -2, 1, and 4.
Step 5 :By analyzing the sign of each factor in the intervals determined by these roots, we find that the derivative is negative in the interval (-2, 1).
Step 6 :Thus, the function $f(x)$ is decreasing on the interval $\boxed{(-2,1)}$.