Let's denote the number of workers initially hired as \( n = 5 \), and the total work required to complete the job as \( W \).
Let \( r \) be the work rate of one worker (i.e., the amount of work one worker can do in one day). Then the work rate of \( n \) workers is \( nr \), and the time it takes \( n \) workers to complete the job is:
\[
\text{Time} = \frac{W}{nr}
\]
### Step 1: Set up the equation for one additional worker
If one additional worker is hired, the total number of workers becomes \( n+1 \), and they can complete the job 12 days earlier. Therefore, the time it would take \( n+1 \) workers to complete the job is:
\[
\frac{W}{(n+1)r} = \frac{W}{nr} - 12
\]
We now substitute \( \frac{W}{nr} \) with the initial time \( T \):
\[
\frac{W}{(n+1)r} = T - 12
\]
Equating the two expressions for time:
\[
\frac{W}{(n+1)r} = \frac{W}{nr} - 12
\]
### Step 2: Solve for \( T \)
Substitute \( T = \frac{W}{nr} \):
\[
\frac{W}{(n+1)r} = \frac{W}{nr} - 12
\]
Multiply both sides by \( (n+1)r \) to clear the fractions:
\[
W = \frac{W(n+1)}{n} - 12(n+1)r
\]
Simplify and solve for \( T \):
\[
W = \frac{Wn + W}{n} - 12(n+1)r
\]
Simplify the equation:
\[
W = W + 12nr - 12r = 12nr \quad \Rightarrow \quad T = \frac{W}{nr} = 12
\]
Now, let's calculate the number of additional workers needed to complete the job 32 days earlier.
### Step 3: Set up the equation for \( k \) additional workers
If \( k \) additional workers are hired, the total number of workers becomes \( n + k \), and they can complete the job 32 days earlier. The equation for time is now:
\[
\frac{W}{(n+k)r} = T - 32
\]
Using the expression \( T = \frac{W}{nr} \):
\[
\frac{W}{(n+k)r} = \frac{W}{nr} - 32
\]
Multiply both sides by \( (n+k)r \):
\[
W = \frac{W(n+k)}{n} - 32(n+k)r
\]
Simplify the equation:
\[
W = W + 32(nr) - 32r = 32nr
\]
Now solve for \( k \):
\[
\frac{W}{nr} - \frac{W}{(n+k)r} = 32 \quad \Rightarrow \quad k = 8
\]
### Final Answer
Thus, 8 additional workers should be hired to complete the job 32 days earlier. The correct answer is:
\[
\boxed{8}
\]
