If you are really talking about 2 by 2 matrices that have polynomials of degree 2 or less as entries, then "

" makes no sense. And

is not a basis for any subset of that.

I suspect you mean simply "the vector space of all polynomials of degree 2 or less".

In that case, any such "vector" can be written as

and any vector in the "orthogonal complement of

" must satisfy

and

. Do those integrals to get equations for a, b, and c. Since those are two equations, you will probably be able to solve for two of them in terms of the third. Write replace those two by their expression in terms of the third to get

in terms of just one of a, b, or c. Finally, factor that a, b, or c out leaving just a number times a polynomial. That polynomial will be the single vector in the basis.