determine Maclaurin development of order 3 with residual term with o notation e^sinx

My main problem is that i don't understand the rules of O notation

1. e^sinx ... u= sinx

we get e^u

2. Develeopment of e^x = 1 + x + x^2/2 + (O(x^4))

Develeopment of sinx = x - 1/6 x^3 + O(x^5)

3. e^sinx = e^x-x^3/6 * e^(O(x^5))

= e^x * e^-x^3/6 * e(O(x^5))

I could need some help finish this off

secondly I don't understand why we put an O notation of O(x^5)) here out of a sudden, we had an ordo of 4 as well didnt we?

Would be grateful if someone could explain all the Ordo (O notation) rules so that i can start working on problem self without using hints like the one above. All that we have is a booklet handout with problems + a very bad book that does not cover the topic well.

Thanks alot