How do you work with these imagnary numbers?

(3 + 2i) ( 4 + i)  =

4(3 + 2i) + i  ( 3 + 2i)  =

12 + 8i + 3i + 2i^2   =

12 + 11i + 2(-1) 

10 + 11i

(5 -  2i)^3  =

(5 - 2i) ( 5 - 2i) ( 5 - 2i) =

(25 - 5*2i - 5*2i + 4i^2) ( 5 - 2i) =

(25 - 20i + 4(-1) ) ( 5 - 2i) =

(21 - 20i) ( 5 - 2i) =

(105  - 100i - 42i + 40i^2 ) =

(105 - 142 i + 40(-1) ) =

 65 - 142 i 

(a + bi) ( a - bi)  =

a^2  + abi - abi - b^2i^2 =

a^2 - b^2(-1) =

a^2 + b^2