f(a+b)=f(a)+f(b)+ab

 

f(0)=f(0)+f(0)+0 =>f(0)=0

 

f(0)=f(1+(-1))=f(1) + f(-1) -1=0 =>f(1) + f(-1) =1

 

similarly f(2) + f(-2) =4

 

and f(a) + f(-a) =a^2

 

if f(x) is odd f(a) = - f(-a) =>4=0

 

therefore f(x) is not odd

 

if f(x) is even then f(a) = f(-a)

 

=>2f(a) = a^2

 

therefore f(a)=(a^2)/2

 

we can see that f(x)= X²/2 satisfies the given condition ( I donot mean to say that functions are either even or odd but you get the answer )