Basis of 4th degree polynomial linear space that doesn't contain 2nd or 3rd degree polynomials In K4[x], the linear space of the polynomials of degree 4 or less with coefficients in in K, determine:

(a) A basis that doesn't contain 2nd or 3rd degree polynomials 

(b) If there exists a basis that doesn't contain 4th degree polynomials

(c) A supplementary subspace of the one generated by the polynomials
p(x)=−1+2x+x^2 y q(x)=1+x+x^2 

Suggestion: use coordinate row (or column) matrices
The basis for a 4th-degree polynomial is {1,x,x^2,x^3,x^4}. So I think the answer to part (a) is {1,x,x^4}.
For part (b) there is not.
The basis for a 4th-degree polynomial is {1,x,x^2,x^3,x^4}. So I think the answer to part (a) is {1,x,x^4}.
For part (b) there is not.