I do not have your Edward b. 1805 in my data base but you may want to start with the first Alexander to that area and work your way forward.
Anthony Alexander "humbly Sheweth that he hath right to 200 acres of Land by the Importacon of himselfe Ann his wife, Anthony his son & Jno Mason his servt, wch he is ready to prove & prays a Cert & shall pray. w. Glover Cl Cur." (from NC Historical & Genealogical Records)
"North Caroline Headrights - A List of Names, 1663 - 1744" pg. 30
p. 225 {CCR 102} Anthony Alexander proves his right to two hundred acres of land by the Importation of himself, An his Wife Anthony his son and Jno. Mason (General Court, 28 July 1698)
Anthony and Ann appear to have had 19 children. You can find his will and much more on the Alexanders at
http://www.ncgenweb.us/tyrrell/TYRRELL.HTMSo far as I know Anthony Alexander was the progenitor of all the Alexanders in that area of NC.
Jean