If $$4a + 2b + c = 0$$    then the equation $$3a{x^2} + 2bx + c = 0$$     has at least one real root lying between :

Solution :
$$3a{x^2} + 2bx + c = 0\,\,\,\,\, \Rightarrow a{x^3} + b{x^2} + cx = 0\,\,\left( {{\text{on integration}}} \right)$$
Clearly $$x=0,\,2$$   are the roots of $$a{x^3} + b{x^2} + cx = 0\,\,\,\,\,\left( {\because 4a + 2b + c = 0} \right)$$
$$\therefore $$ the derived equation $$3a{x^2} + 2bx + c = 0$$     will have a root lying between 0 and 2.