55*45+66*54+77*63+88*72
=(50+5)(50-5) + (60+6)(60-6) + (70+7)(70-7) + (80+8)(80-8)
=50^2 - 5^2 + 60^2 -6^2 +70^2 -7^2 +80^2 -8^2
= (5^2+6^2+7^2+8^2)*(100-1)
= 17400-174
= 17226