Using the method $a^2 - b^2 = (a + b)(a - b)$

$\large 9999^2 + 1^2 = 9999^2 - 1^2 + 1^2 + 1^2$

$\large= (9999+1)(9999-1) + 2$

$\large = 10000\times 9998 + 2$

$\large = 99980000 + 2$

$\large = \boxed{99980002}$

Notice that:

$99^2 = 9801\\ 999^2 = 998001\\ 9999^2 = 99980001$

Therefore,

$9999^2 + 1^2 = 99980001 + 1 = \boxed{99980002}$

Alternate method: use $(a-b)^2 = a^2 - 2ab + b^2$

$9999^2 + 1^2\\ =(10000-1)^2 + 1^2\\ =100000000 - 20000 + 1 + 1\\ =99980000 + 2\\ =\boxed{99980002}$

Relevant wiki: Difference Of Squares

As the tips given, use the method a^2 − b^2 = (a + b)(a − b), therefore

(9999)^2 + 1 =(9999)^2 + 1^2 − 1^2 + 1^2
*Do this in order to use the mehod difference of square

=(9999^2 − 1^2) + (1^2 +1^2)
*Move the places to get a^2 − b^2

= (10000 x 9998 ) + 2
*a^2 − b^2 = (a + b)(a − b)

= 99980002