Question

Find a pair of conjugates with a product of 5

Answer 1

(a-bi)(a+bi) = a^2 + b^2
so what two small perfect squares do you know that add up to 5?

Now if you are sticking to real numbers, and mean quadratic conjugates, then
(a+b)(a-b) = a^2 - b^2
so what two small perfect squares do you know that differ by 5?