Question

give five examples of linear transformations T:p2 -->R4

Answer 1

How about:
T(ax^2+bx+c)->(a,b,c,b^2-4ac)

Answer 2

for real?

Answer 3

Yes, as long as a,b, c and x are real!
In your question, p2 stands for polynomial of degree 2, if I am not mistaken.
You can make up anything you want, but that's the one that came to my mind.