(a): {1, x^2, x^2 – 2}

(a) Yes.

a + bx + cx² = (a+2c) (1) + b (x) + c (x²-2)

(b) Yes.

a + bx + cx² = ( a/2 ) (2) + b (x) + c (x²) + 0 (2x+3)

(c) Yes.

a + bx + cx² = (2b-c-a)(x+1) + (a+b-c)(x+2) + c(x²-1)

(d) No. It is not big enough (to span P3 you need 3 elements).

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Response to “additional details” — to find out if any three elements p(x) q(x) r(x) form a spanning set, solve the equations:

a + bx + cx² = Ap(x) + Bq(x) + Cr(x).

If solutions for A,B,C exist (with respect to a,b,c), then this is a spanning set, since you can achieve any a,b,c set using the correct choice of A,B,C.

In the case where you had four things, you can do it just with three, since three things is enough to span P3, or you can find solutions to the equation with an extra summand Ds(x). If so, there will possibly be more than one solution for A,B,C,D.