In an example from the book, the polynomials $\displaystyle x^2+3x-2 \ , \ 2x^2+5x-3 \ , \ -x^2-4x+4 $ generate $\displaystyle P_{2}(R)$ since they are all in the set and each polynomial in $\displaystyle P_{2}(R)$ is a linear combination of these three.

What I don't understand is the what is $\displaystyle P_{2}(R)$ and how do you know that all poly in the set are linear combination of these three?

thank you.