x(x+a)(x+2a)(x+3a) =(x2+ax)(x2+5ax+6a2) =x4+ax3+5ax3+5a2x2 +6a2x2+6a3x =x4+ax(x2+5x2+5ax 6ax+6a2) =x4+ax(6x2+11ax+6a2)⋅⋅⋅⋅⋅⋅⋅(i) Now, for terms to be perfect square, (x+y)2(x+y)2 =(x2+2xy+y2)(x2+y2+2xy) =x4+2x3y+x2y2+x2y2 +2xy3+y4 +2x3y+4x2y2+2xy3 =x4+xy(4x2+6xy+4y2)+y4... (ii) On comparing Eqs. (i) and (ii), as y=a So, a must be added to make it a perfect square.