If(x) =ax^4=bx^3+cx^2+dx+e(a,b,c,d,eER0) such that f(1)+f(2) = f(2)+f(3) =f(3)+f(4)=0, then equation f(x)=0 have1all roots non real2. two real and two non real roots3 all four roots are real & equal4 all four roots are real and distinct

Let f(1)>0
then f(2)<0
if f(2)<0 then f(3) >0
if f(3)>0 then f(4) <0

So between 1 and 2 the graph will cut x axis once
similarily between 2 and 3 once
and between 3 and 4 once
so 3 real rools we will get
But if there was any complex root it will exits in pais
So fourth root will also be real
So, All 4 real roots