# Determine tranformation matrix of linear transformation

A linear transformation between polynomial spaces $f: P_2(\mathbb{R}) \to P_2(\mathbb{R})$ is given by $$f(p(x))=3 \cdot p(1)-x^2 \cdot p(0)+(x-1) \cdot p'(1)$$ Determine the transformation matrix with respect to the monomial basis $(1,x,x^2)$

I tried to approach it this way:

The general form of these polynomials is $ax^2+bx+c$. We have $$P(1)=a+b+c \text{ and } P(0)=c \text{ and } P'(1)=2a+b$$