Note that 0.58 is only a little bigger than 0.5568 so the quotient is going to be close to 1. This tells us where to put the decimal point. We can ignore the decimal points for now. Now ask: can the problem be simplified? In this case it can. Consider the numbers without the decimal points by multiplying each by 10000:
5800/5568. The numbers are now the same sort of size. Now divide by common factors. Start with 8:
725/696. 725=25×29 and 696=8×3×29=24×29, so 29 is a common factor, leaving:
25/24 as the fraction which is 1+1/24. 1/24=¼×⅙. ¼=0.25, so divide 0.25 by 6=0.0416666...
So the quotient is 1+0.0416666...=1.0416666... etc.
But if you would really like to do the long division, here it is:
1.04166...
5568)5800.00
-5568
23200
-22272
9280
-5568
37120
-33408
3712 this is the same remainder as before so we have a recurring decimal.