Profitc=v-cs-cv-r (α p+cs)+(1-e)*(α p+cs)-(1-e) θ-k*e*e

Profitc

Simplify[-cs-cv-(e^2) k+v+(1-e) (cs+p α)-r (cs+p α)-(1-e) θ]

D[Profitc,e]

Solve[-cs-2 e k-p α+θ==0,e]

{{e->((-cs-p α+θ)/(2 k))}}
e:=((-cs-p α+θ)/(2 k))

Profitc=v-cs-cv-r (α p+cs)+(1-e)*(α p+cs)-(1-e) θ-k*e*e

Simplify[Profitc]

Factor[(((cs^2)-4 cv k+4 k v+4 k p α-4 k p r α+(p^2) (α^2)-4 k θ-2 p α θ+(θ^2)-2 cs (2 k r-p α+θ))/(4 k))]

e:=((-cs-p α+θ)/(2 k))

e

r=β(1-e)

Simplify[r]