First, it is important to understand that the function changes its sign whenever it crosses zero. And this occurs at the roots of the function.
Therefore, the first step is to determine the roots of the function. The roots are the possible values of x that will make the function equal to zero. The equation to find the roots is given as follows:
(3 - x)(x + 1)(x + 5) = 0
By setting each factor to zero we find the roots to be at x = -5, x = -1, and x = 3.
Now we need to determine the intervals where the expression is less than or equal to zero. We know that the function changes its sign at the roots, x = -5, x = -1 and x = 3. So we break up the real number line into the intervals based on these roots:
...] - (-infinity, -5), (-5, -1), (-1, 3), and (3, infinity)
To determine if the function is positive or negative over these intervals, we can select a test point within each interval and evaluate the function. But due to the problem's condition, we don't need to check if the function is positive or negative. We just need to check if the function is less than or equal to zero.
So our solution is that for the inequality (3 - x)(x + 1)(x + 5) <= 0, the solution set is the intervals (-infinity, -5), (-5, -1), (-1, 3), and (3, infinity), where each interval is inclusive of its endpoints (because the inequality is less than or equal to zero, rather than strictly less than zero).