I am assuming P is a polynomial since we are mapping on . Unless your book notates this differently, I am almost positive these are polynomials; however, different books notate differently. For instance, mine says are polynomial of degree 1 or less. Thus, the 2 in is the amount of elements {1, x} but some books refer to as degree two or less, { , , }. What does your book say?
The "set of all 2nd degree polynomials" does not form a vector space. As dwsmith said, is the set of all polynomials of degree 2 or less.
Random Variable showed how to represent this transformation as a matrix- worth knowing for itself.
But for this problem, you don't have to do that. As dwsmith said, if , then . If is an eigenvalue for T, we must have some a, b, c, not all 0, such that . Since that must be true for all x, we can set corresponding coefficients equal: , , and .
If a is not 0, then , if b is not 0, then , and if c is not 0, then .
That is, 0 is an eigenvalue with eigenvectors any multiple of 1, 1 is an eigenvalue with eigenvectors any multiple of x, and 2 is an eigenvalue with eigenvectors any multiple of , exactly what Random Variable got.