Integrate[(E^(-(1-a-b x-c(((x-m))^2)))),{x,-Infinity,+Infinity}]

Integrate[x (E^(-(1-a-b x-c(((x-m))^2)))),{x,-Infinity,+Infinity}]

Integrate[(((x-m))^2) (E^(-(1-a-b x-c(((x-m))^2)))),{x,-Infinity,+Infinity}]

Solve[{(((E^(-1+a-((b^2)/(4 c))+b m)) (Sqrt[π]))/(Sqrt[-c]))==1,(((E^(-1+a-((b^2)/(4 c))+b m)) (b-2 c m) (Sqrt[π]))/(2 (((-c))^(3/2))))==m,((((b^2)-2 c) (E^(-1+a-((b^2)/(4 c))+b m)) (Sqrt[π]))/(4 (((-c))^(5/2))))==(s^2)},{a,b,c}]