Let \( d \) be the number of days it would take for the original 5 workers to complete the job. The total amount of work can be expressed in terms of man-days, which is the product of the number of workers and the number of days they work. Therefore, the total work required for the job is given by:
\[ \text{Total work} = 5d \text{ man-days} \]
When 1 additional worker is hired (making it 6 workers), the time taken to complete the job decreases by 12 days, so the new time to finish the job is:
\[ d - 12 \]
The equation for the work done by 6 workers in this new time is:
\[ \text{Total work} = 6(d - 12) \text{ man-days} \]
Since the total work remains constant, we set the two expressions for total work equal to each other:
\[ 5d = 6(d - 12) \]
Expanding the right side gives:
\[ 5d = 6d - 72 \]
Rearranging this equation yields:
\[ 72 = 6d - 5d \]
Therefore, we find:
\[ d = 72 \]
Now we know that the time it takes for 5 workers to complete the job is 72 days.
Next, when we want the job to be completed 32 days earlier, the new target completion time becomes:
\[ 72 - 32 = 40 \text{ days} \]
Let \( n \) be the total number of workers we need to hire to complete the job in 40 days. In terms of man-days, we have:
\[ n \times 40 = 5d \]
Substituting \( d = 72 \) gives us:
\[ 5 \times 72 = 360 \text{ man-days} \]
This means we can express the relationship as:
\[ n \times 40 = 360 \]
Solving for \( n \):
\[ n = \frac{360}{40} = 9 \]
Since we originally had 5 workers, the number of additional workers needed is:
\[ 9 - 5 = 4 \]
Thus, the number of additional workers that should be hired to complete the job 32 days earlier is:
\[ \boxed{4} \]