Write x^2-2x-3=0 in the form (x-a)^2=b, where a and b are integers

This is worded in a way meant to throw you off, i think, but all it's asking you to do is complete the square! it will give you the squared binomial on the left, then your leftover integer on the right (b).

x² - 2x - 3 = 0  ... add 3 to both sides
x² - 2x + __ = 3  ... the empty space is where you'll need to complete the square. this is done by dividing your "middle" coefficient (-2) by 2, then squaring it. -2/2 is -1 ... -1² is 1. therefore:

x² - 2x + 1 = 3 + 1

you have to balance it by adding the 1 to both sides. what you do to one side, you have to do to the other. the next step is to factor the left side:
(x - 1)(x - 1) = 4

rewrite it as (x - 1)² = 4  ... this is your answer.