1) x+3/3 = x+1. You can cancel the 3's. However, you probably meant (x+3)/3 which = x/3 + 1. You can cancel the 3's but only if you also divide x by 3
2) no cancellation is possible. You probably mean x-3/(x+3), or (x-3)/(x+3), but either way, as x-(3/x) + 3 or however you put in the parentheses, no cancellation is possible. what you could do with (x-3)/(x+3) is multiply by x-3 or by x+3 to get (x-3)^2/(x^-9) or (x^2-9)/(x+3)^2
Or just do long division, divide x+3 into x-3 to get 1 with remainder -6 or 1 -6/(x+3)
Cancellations require division that divides evenly with no remainder
third case: 3(x+1)/(3x+1) = (3x+3)/(3x+1) You need a common factor in both numerator and denominator to cancel. 3(x+1)/3(x+1/3) has a common 3 that can be canceled to get (x+1)/(x+1/3). But if you just did long division dividing 3x+1 into 3(x+1) you get 1 with remainder 2. It doesn't divide evenly
4th case: 3x divided by 4/3. If you flip the divisor upside down you can change division to multiplication.
3x divided by 4/3 = 3x times 3/4 = 9x/4. Division leads to a remainder. It doesn't divide evenly. Or 3x and 4/3 have no common factor. 3 is not a common factor. 3 is a factor of 3x, but not of 4/3. 1/3 is a factor of 4/3, but 3 is not a factor of 4/3