We rationalize the denominator,
x=√5−2√5+2×√5−2√5−2
x=5+4−3√55−4
x=9−4√51.
Then, x2=(9−4√5)(9−4√5)
=81+16×5−72√5
=161−72√5 .
We rationalize the denominator,
y=√5+2√5−2×√5+2√5+2
y=5+4+4√55−4
y=9+4√51.
Then, y2=(9+4√5)(9+4√5)
=81+16×5+72√5
=161+72√5 .
Now, x=√5+2√5−2
and y=√5−2√5+2.
Then, xy=√5+2√5−2×√5−2√5+2
xy=1 .
Therefore, x2+y2+xy=161−72√5+161+72√5+1=323.