I have attached the problem:

We are dealing with polynomials and we have a linear transformation $\displaystyle F:P_2 \rightarrow P_3$ , meaning we are transforming a polynomial of degree 2 to a polynomial of degree 3.

An intellectual guess for the basis of the Image of the linear transformation would be $\displaystyle 1,x,x^2,x^3$, however I'm not too sure.

I would like some guidance on how to encounter questions like these about Images and Kernals.

Thank you.