looking at various solutions I'm pretty sure you mean , not what you have written.

a linear polynomial is

a quadratic polynomial is

first step should just be plugging these in and confirming that they are solutions.

For example for the linear polynomial

so plugging these in we get

so any linear polynomial of the form ax+a is a solution. This can be thought of as a(x+1), i.e. (x+1) serves as a basis vector of the solution space.

Now repeat this with the quadratic. and remember superposition, and then you'll start to understand what they are after in part 2 of the problem.