Problem:
Simplify
bx+aybx(a2x2+2a2y2+b2y2)+ay(a2x2+2b2x2+b2y2)​
Answer Choices:
A. a2x2+b2y2
B. (ax+by)2
C. (ax+by)(bx+ay)
D. 2(a2x2+b2y2)
E. (bx+ay)2
Solution:
Let N be the numerator in the expression to be simplified. The answers given suggest that N is divisible by bx+ay. If the differing middle terms in the two parenthesized expressions in N, namely, 2a2y2 and 2b2x2, were not there, divisibility would be obvious. So let us separate the middle terms off. We obtain
N​=bx(a2x2+b2y2)+ay(a2x2+b2y2)+bx(2a2y2)+ay(2b2x2)=(bx+ay)(a2x2+b2y2)+2a2bxy2+2ab2x2y=(bx+ay)(a2x2+b2y2)+2abxy(ay+bx)​
Thus
bx+ayN​=a2x2+b2y2+2abxy=(ax+by)2