Is there a basis of the set of polynomials with degree less than $4$ such that none of the basis polynomials have degree 1? 

I tried $\{1,x^2,x^3-x,x^3+x\}$ and showed they are linearly independent but I don't think they span the whole set as in $\{a,b,c,d\in \mathbb{R}:a+(d-c)x+bx^2+(c+d)x^3\}$, the coefficients of $x$ and $x^3$ cannot be chosen freely.