We can solve this together! We're beginning with 385382 and removing 18966 from it.

Step 1: We'll start by subtracting the digit 6 and the borrow 0 from 2 in column 1 and get -4.
We add -4 to 10 and get 6 as the first digit of the result.

Step 2: We'll start by subtracting the digit 6 and the borrow 1 from 8 in column 2 and get 1.
1 is the next digit of our result.

Step 3: We'll start by subtracting the digit 9 and the borrow 0 from 3 in column 3 and get -6.
We add -6 to 10 and get 4 as the next digit of the result.

Step 4: We'll start by subtracting the digit 8 and the borrow 1 from 5 in column 4 and get -4.
We add -4 to 10 and get 6 as the next digit of the result.

Step 5: We'll start by subtracting the digit 1 and the borrow 1 from 8 in column 5 and get 6.
6 is the next digit of our result.

Step 6: We'll start by subtracting the digit 0 and the borrow 0 from 3 in column 6 and get 3.
3 is the next digit of our result.

So, 385382 - 18966 = 366416.