Question

Why we write ax+b in place of r(x) when we have to find remainder in p(x)=[(x-α)(x-β]q(x)+r(x)

Answer 1

The given polynomial p(x) is of degree 2 ie highest power of p(x) is 2. This is because (x-α)(x-β) will give the x^2 term.
So, the remainder will obviously be an expression with degree < 2. Hence we assume the remainder r(x) is a linear polynomial and take the general form of r(x) as ax+b.

Hope this helps :)