We can make up any rule we want. As long as we are consistent about it.
So what is 5+4x3+2?
We could make up a rule that: 1) You always do it strictly left to right
So 5+4x3+2 = 9x3+2=27+2=29.
Or we could make up a rule that: 2) You always do addition first
So 5+4x3+2 = 9x5 = 45.
Or we could make up a rule that: 3) You always do multiplication first
So 5+4x3+2 = 5+12+2 = 19.
Or we could make up a rule that: 4) You always go right to left
So 5+4x3+2 = 5+4x5=5+20=25.
So which rule is best?
Well, for many reason 3 is best and 1 and 4 are the worst. But really we could get by with any one provided once we pick one we stick with it.
For many reasons "Multiplication first; Addition second".
What about brackets and parenthesis? Well, the entire reason we have brackets and parenthesis is to tell us to do things first. The are precisely used when the normal rules are not what we want to do so we put them in to indicate something must be done first.
Seriously, if we had a rule that we had to do parenthesis last you can see that that wouldn't work. How could we express "3 times the result 4 plus 5" if we can have any way to say "add the 4 and 5 first". "Add the 4 and 5 first" is what 3x(4+5) means.
So why do we do multiplication first, then addition? Or for that matter powers first, then multiplication, and addition?
Well, I think it's because of "grouping". When we add things we are grouping but sets of units. 3 + 5 is really "3 ones grouped with 5 ones makes 8 ones". When we multiple we are grouping by big factors rather than small units. 3x4 + 5x6 means "we have a set of 3 fours and a set of 5 sixes; that combines and we have 12 and 30 and we combine them by the units to make 32". I don't know. That seems to me the most natural way to do it. In my opinion anyway....
So 3x(4+5) means "okay, first we specifically group the 4 and 5 and then we take a set of 3 of the results of 9. Three 9s is 27".
And powers are in even larger grouping.
Okay... so what about subtraction and division.
Well, addition/subtraction are inverses. 5 - 3 means ? + 3 = 5. Or more algebraically 5 + [-3] where [-3] is the number that takes 3 away. Basically subtraction and addition are the same level of grouping. It doesn't really matter which you do first. I think the BODMAS mnemonic fails as with 3 -4 +5 you definitely dont want to add the 4+5 to get 3-4+5 = 3 - 7 before you subtract. You really want to consider subtraction is adding negative numbers 3 - 4 + 5 is 3 + [-4] + 5 and now it's just addition in any order.
And likewise division, $8\div 4$, is the inverse of multiplication. $8 \div 4 = 8 \times \frac 14$.
So... no. The distinction between addition and subtraction is not as important as all that. BUT do be careful. If you get cavilier mistakes will happen.
Anyway, BODMAS is just a memory aid. It's not actually a mathematical rule.