For many, the instinctive reaction may be to try to find the values of X and Y. However, since the problem never asks for those values, we should wonder: Is it possible to find the sum X² + Y² without actually finding X and Y? The answer is yes.

Beginning with X²Y + XY² + X + Y = 99, we rearrange the terms to arrive at:

XY(X + Y) + (X + Y) = 99
(XY + 1)(X + Y) = 99

Since XY = 10,

11(X + Y) = 99
X + Y = 9

Now take this last equation and square it, giving:

X² + 2XY + Y² = 81
X² + Y² + 2(10) = 81
X² + Y² = 81 - 20 = 61