Let V be the R-space of real polynomials of degree at most 3.

For p(X) in V, define

Tp(X)=1/6(19-6x+7x^2+32x^3-15x^4-6x^5)p'''(X)+(6-7x-9x^2+5x^3+x^4)p''(X)+(7-5x^2+2x^3)p'(X)+(10-6x^2)p(X).

I need to find a basis for each of the kernel ker(T) of T, and the image (or range) Im(T) of T (T is a linear operator on V).

Any ideas?