You need the LCM of 12 and 8 first. 12=2*2*3 and 8=2*2*2. The LCM needs to contain three 2's and one 3: so LCM=2*2*2*3=24. We can see that 2*2*3 and 2*2*2 can be found in 2*2*2*3.
So we now work out how many times 12 (=2*2*3) goes into 24 (2*2*2*3). When we remove 2*2*3 from 2*2*2*3 we're left with 2, so that's how many times 12 goes into 24. Multiply the numerator by this number and we get 10, so 5/12 is the same as 10/24. Now we do the same with the other fraction: when we divide 8 into 24 we get 3, so we multiply 7 by 3=21. The second fraction is the same as 21/24.
The problem is now 10/24-21/24. Or we can write this (10-21)÷24. The result of the top is -11 so the answer is -11÷24 or -11/24 as a fraction.