Determining which sets are spanning sets for P3? I have a list of sets:
(a): {$1, x^2, x^2 - 2$} 
(b): {$2, x^2, x, 2x + 3$} 
(c): {$x + 2, x + 1, x^2 - 1$} 
(d): {$x + 2, x^2 -1$}
I am supposed to determine which of the following are spanning sets in $P_3$.
My attempted solution:

(A): Ax^2 + bx + c = A(1) + B(x^2) +C(x^2-2)

 Ax^2 + bx + c = (A-2C) + Bx^2 + C(x^2)

This is pretty much as far as I've got and I am not exactly sure what I'm doing or solving for. Any clarification would be wonderful. 

I found b) and c) are the spanning sets. 
For b) You only need to show that you can find reals m, n, p such that:
ax^2 + bx + c = 2m + nx^2 + px. You don't need 2x + 3 since 2x + 3 = 2x + (3/2)*2 is in the span{2,x}. So choose n = a, p = b, and m = c/2.
For c) again ax^2 + bx + c = m(x+2) + n(x+1) + p(x^2 - 1). Choose p = a, and :
m = b - a - c, and n = a + c. 
For a) x^2 + 2x + 1 is not in the span{1, x^2, x^2 - 2}.
For d) ax^2 + bx + c = m(x+2) + n(x^2 -1) ==> n = a , m = b, c = 2m - n = 2b - a. So you only need to find a quadratic polynomial like x^2 + x - 5, then it is not in the span{x+2, x^2 - 1}.
