I have this question on an assignment, and I live off-campus and office hours are hard to go to.
I was able to complete part (a), by showing that the inner product of the 2 vectors is equal to 0. For part (b) I have no idea what to do. I know that the standard basis for $P_2(R)$ are $\{1,x,x^2\}$ but how do I come up with an orthogonal basis that contains $S$?
Since $S$ is orthogonal we need to find a vector that is orthogonal to $S$ since the dimension of $P_2(R)$ is three. Starting with $ax^2+bx+c$ ends in $-10x^2+99x+10$.