Question 31503
I DONT KNOW YOUR COURSE OF STUDY AND BACK GROUND KNOWLEDGE.I HOPE YOU KNOW CALCULUS AND ROLLE'S THEOREM.I AM ANSWERING ON THAT BASIS
CONSIDER F(X)=AX^3+BX^2+CX.....
WE HAVE 
F(0)=A*0^3+B*0^2+C*0=0
F(1)=A*1^3+B*1^2+C*1=A+B+C=0..AS GIVEN....SO AS PER ROLLE'S THEOREM 
WHEN A CONTINUOUSLY VARYING FUNCTION HAS 2 ZEROES AT BOTH ENDS OF AN INTERVAL...THEN ITS DERIVATIVE WILL HAVE A ZERO AT ONE POINT WITHIN THE INTERVAL.HENCE
DF/DX=3A*X^2+2B*X+C=0 BETWEEN X=0 AND X=1   
SO B IS THE ANSWER