We're dividing 259837 by 45 
We're looking to find how many times 45 goes into 259837.

Advancing to step 1:
The number 45 fits into 2 0 times, leaving a remainder of 2.
Write down 0 as next digit of the result. 
Result so far: 0.0
Subtracting 0 from 2 leaves us with 2.
Bring next digit (5) of the dividend behind the 2 and repeat the process: 25 / 45

Moving on to step 2:
If we divide 25 by 45, we get 0 and a remainder of 25.
The next digit of our result is 0. 
Result so far: 0.0
Deduct 0 from 25 and we're left with 25.
Take the next digit (9) from the dividend and append it to 25, then repeat: 259 / 45

Moving on to step 3:
45 can be fit into 259 5 times, resulting in a remainder of 34.
The number 5 becomes the next digit in our result. 
Result so far: 5.0
The remainder is 34 after subtracting 225 from 259.
Include the next digit (8) from the dividend after 34, then repeat: 348 / 45

Going ahead to step 4:
If we divide 348 by 45, we get 7 and a remainder of 33.
Write down 7 as next digit of the result. 
Result so far: 57.0
If we take 315 away from 348, we end up with 33.
Append the next digit (3) from the dividend to 33 and continue with: 333 / 45

Let's proceed to step 5:
333 divided by 45 is 7 with a remainder of 18.
Record the quotient 7 as the next digit in the result. 
Result so far: 577.0
Subtracting 315 from 333 leaves us with 18.
Fetch the next digit (7) from the dividend, attach it to 18 and continue: 187 / 45

Step 6:
45 goes into 187 4 times with a remainder of 7.
Write down 4 as next digit of the result. 
Result so far: 5774.0
The remainder is 7 after subtracting 180 from 187.
Since there are no more digits in the dividend, we add a zero to the remainder making it 70, and continue the process.

On to step 7:
45 goes into 70 1 times with a remainder of 25.
Record the quotient 1 as the next digit in the result. 
Result so far: 5774.1
Subtract 45 from 70 to get 25.
Since there are no more digits in the dividend, we add a zero to the remainder making it 250, and continue the process.

Going ahead to step 8:
250 divided by 45 is 5 with a remainder of 25.
The number 5 becomes the next digit in our result. 
Result so far: 5774.15
Subtract 225 from 250 to get 25.
Since there are no more digits in the dividend, we add a zero to the remainder making it 250, and continue the process.

Going ahead to step 9:
If we divide 250 by 45, we get 5 and a remainder of 25.
Use 5 as the next digit of our solution. 
Result so far: 5774.155
Deduct 225 from 250 and we're left with 25.
Since there are no more digits in the dividend, we add a zero to the remainder making it 250, and continue the process.

In conclusion, the division gives 5774.155 with a balance of 0.0025.