If possible, find a basis n = {n1, n2, n3} of P2(R) such that

[2 + 5x + 4x^2]n = [1,2,3] , [1 + x + x^2]n = [4,1,2] and

[x + x^2]n = [3, -5, 1]

Basically, from what I understand, we have Ax = b.

I can find a basis, for one of these, but not all three.

like, [a,b,c] * [2,5,4]^t = [1,2,3]^t

[d,e,f]

[g,h,i]

0, 1, -1

1, 0, 0

-1, 1, 0 so, [0,1,-1], [1,0,1] [-1,0,0] would be a basis for the first one, but the others... I'm not sure...

Thank you