algebra help

[JaysFan31]

1. The problem statement, all variables and given/known data
I'm working on change-of-basis matrices and vector spaces.

I have B=(1+x+x^2+x^3, 1+x+x^2, 1+x, 1) and C=(1+x^2, 1+x+2x^2, 1+4x+5x^2+x^3, -2+2x-x^2+5x^3).

Also, p(x)=3-8x+2x^2-6x^3.

I need to find [p]subscriptB and [p]subscriptC.

I need to show that the change of basis matrix from C to B times [p]subB=[p]subC and it doesn't work. I know the change of basis matrices.


2. Relevant equations
Nothing relevant that I can think of. It's probably what I'm missing. What exactly is the equation/formula for [p]subscriptB and [p]subscriptC?


3. The attempt at a solution
Here's my work for finding [p]subscript B:
I have the equation p(x)=3-8x+2x^2-6x^3 and B=(1+x+x^2+x^3, 1+x+x^2, 1+x, 1).
I said a(1+x+x^2+x^3)+b(1+x+x^2)+c(1+x)+d=3-8x+2x^2-6x^3.
I solved for a,b,c,d and got:
a=-6
b+a=2
a+b+c=-8
a+b+c+d=3

Thus, a=-6, b=8, c=-10, d=11.

So I said that [p]subscriptB is the vector:
[-6
8
-10
11]
, but this is obviously not right.

Any help or guidance would be greatly appreciated.