I find this very hard to understand. I think you mean that you want to determine whether or not the subset of the vector space of all polynomials of degree two or less, with the property that p'(1)= p(2), is a subspace.
To show that it is a subspace, you must show that, if p1 and p2 both satisfy that equation, then so does ap1+ bp2 for a, b any numbers.
(ap1'+ bp2')(1)= ap1'(1)+ bp2'(1)= a(p1(2))+ b(p2(2)).
To find a basis, remember that any polynomial of degree 2 or less can be written as . Then so "p'(1)= p(2)" becomes 2a+ b= 4a+ 2b+ c or 2a+ b+ c= 0.We can choose two of those to be whatever we choose and solve for the third. For example, if we take a= 1, b= 0 we get 2+ c= 0 or c= -2. is a basis vector. If we take a= 0, b= 1 we get 1+ c= 0 or c= -1. x- 1 is a We have no restriction on a and, whatever b is, c must be -b. x- 1 is a basis vector.
Those are the choices I would make and the basis I would determine. If we choose a= -1, b= 0, then c= 2 so is a basis vector. If we choose a= 0, b= -1, then c= 1 so is a basis vector. That is the basis your textbook gives.