a = ( 2 + i)^2 / ( 3 + i)
a = ( 4 + 4i + i^2) / ( 3 + i)
a = { 3 + 4i) / ( 3 + i)
a = [ ( 3 + 4i) ( 3 - i)] / ( 9 - i^2)
a = ( 9 + 9i - 4i^2) / 10
a = (13 + 9i) / 10
So
1 + 1/a =
1 + 10 / ( 13 + 9i) =
[ 13 + 9i + 10 ] / ( 13 + 9i) =
( 23 + 9i) / ( 13 + 9i) =
( 23 + 9i) ( 13 - 9i) / ( 169 - 81 i^2 ) =
(299 -90i - 81i^2) / 250 =
( 380 - 90i) / 250 =
(38 - 9i) / 25