Checking if given set forms basis for $P_3(\mathbb{R})$

Question

The question says whether the polynomials $x^3+2x-4,\ x^3+x^2-3x+1, $and $x^3+5$ generate $P_3(\mathbb{R})$? I did the question in a sneaky way as follows. We already know $\{1,x,x^2,x^3\}$ forms the basis for $P_3(\mathbb{R})$, so it's dimension is $4$. Since the given set contains only $3$ elements, it won't generate $P_3(\mathbb{R})$. But I am not satisfied with my argument. How to find those polynomials which won't get generate by the span of the above set. Any hint. Thanks.

Answer 1

One demonstration. Let
$A = x^3+2x-4,$
$B = x^3+x^2-3x+1,$
$C = x^3+5$

You can see the subspace Span{$A,B,C$}∩Span{$1,x,x^2$} is generated by
$D = B-A = x^2-5x+5$,
$E = B-C = x^2-3x-4$

Similarly, the subspace Span{$A,B,C$}$\cap$Span{$1,x$} is generated by
$F = E-D = 2x-9$,
which is clearly smaller than Span{$1,x$}