Let ay4+bxy3+cx2y2+dx3y+ex4 =(ax2+pxy−ay2)(x2+qxy+y2) Comparing the coefficients of similar terms, b=aq−p,c=−pq,d=aq+p and e=−a b+d=2aq and e−a=−2a ad+be=2ap and a+c+e=−pq ∵ (b+d)(ad+be)=(2aq)×2ap=4a2pq Also, −(e−a)2(a+c+e)=−(−2a)2(−pq)=4a2pq ∴ (b+d)(ad+be)=−(e−a)2(a+c+e) ⇒ (b+d)(ad+be)+(e−a)2(a+c+e)=0