Determining a basis for a space of polynomials. Let $V = \mathbb R[x]_{\le 3}$
I have the space of polynomials $U_2 = \{ p = a_0 + a_1x + a_2x^2 + a_3x^3 \in V \mid a_1 - a_2 + a_3 = 0, a_0 = a_1   \}$
I am asked to find a basis, so I proceed by noticing that in $U_2$:
$$a_0 + a_1x + a_2x^2 + a_3x^3 = a_0 (1+x-x^3) + a_2 (x^2 +x^3)$$
So I figure that a basis is $(1+x-x^3,x^2 + x^3 )$ In $\mathbb R^4$ the vectors would be $(1 10 -1), (001 1)$  but the solution gives the vectors $(1110), (-1-1 0 1)$.
What am I doing wrong?
Edit: corrected an error pointed out in the comments.

These are both bases. Your basis is $\{1+x-x^3, x^2+x^3\}$; the solution gives the basis $\{1+x+x^2, -1-x+x^3\}$. But each of these is expressible in terms of the other:
\begin{gather*}
  1+x+x^2 = (1+x-x^3) + (x^2+x^3),\quad -1-x+x^3 = -(1+x-x^3) \\
  1+x-x^3 = -(-1-x+x^3),\quad x^2+x^3 = (1+x+x^2) + (-1-x+x^3).
\end{gather*}
