How to decompose the relation R(a,b,c,d,e) whith following FDs into BCNF?
In your relation schema, there are three candidate keys: ABC, ABE and DE.

Since, for instance, AB → D violates the BCNF, we can decompose the original relation in:

R1(ABD) (with dependency AB → D and candidate key AB), and
R2(ABCE) (with dependencies E → C and ABC → E, and candidate keys ABC and ABE)


this is because we decompose in two relations, AB+ and R - (AB+) + AB.

The second relation is still not in BCNF, since in E → C the attribute E is not a superkey.

So we can apply again this method to decompose R2 in:

R3(CE) (with dependency E → C and candidate key E)
R4(ABE) (with no dependency and candidate key ABE)


Both are in BCNF and the final decomposition is constituted by R1, R3, R4.

Finally, it is worth to note that the following dependencies:

DE → AB
ABC → E


are lost in the decomposition.