What is $T(p(a x^2+b))$ when $T(p(x))=x^2p'(x)$

Let $T$ be linear transformation on $mathbb{P} rightarrow mathbb{P}$ that $T(p(x))=x^2p'(x)$.

1. Then, what is the value of $T(p(ax^2+b))$?

I think it could be $x^2p'(ax^2+b)$ or $2ax^3p'(ax^2+b)$ or even something else.

2. What would be the result when $T$ is on $mathbb{D} rightarrow mathbb{D}$? ($mathbb{D}$ is set of differentiable functions)