compute the sum

You can use difference of squares: a² - b² = (a+b)(a-b)

[101² - 97²] + [93² - 89²]

(101-97)(101+97) + (93-89)(93+89)
(4)(198) + (4)(182)

(4)(198 + 182)    [using the distributive property: ab + ac = a(b+c)]
(4)(380) = 1520

Good job ChowMein...! 

Here is another way to do it: 

$101^2 - 97^2 + 93^2 - 89^2$

$=10201-97^2+93^2-89^2$

$=10201-9409+93^2-89^2$

$=10201-9409+8649-89^2$

$=10201-9409+8649-7921$

$=1520$

(ChowMien's way was much easier though)