T: $R^4$ to P2 defined by T(a1, a2, a3, a4)=(a1+a2)+(a2+a3)x+(a3+a4)$x^2$
B={ (1,1,1,1), (1,1,1,0), (1,1,0,0), (1,0,0,0) }
C=( 1, 1+x, 1+$x^2$ )
Question: Find the matrix representation relative to the basis B and basis C
Attempt:
T(1,1,1,1)=2+2x+2$x^2$=a(1,1,1,1)+b(1,1,1,0)+c(1,1,0,0)+d(1,0,0,0)
T(1,1,1,0)=2+2x+$x^2$=a(1,1,1,1)+b(1,1,1,0)+c(1,1,0,0)+d(1,0,0,0)
T(1,1,0,0)=2+x=a(1,1,1,1)+b(1,1,1,0)+c(1,1,0,0)+d(1,0,0,0)
T(1,0,0,0)=1=a(1,1,1,1)+b(1,1,1,0)+c(1,1,0,0)+d(1,0,0,0)
After solving for a,b,c,d I get the matrix
$\begin{bmatrix}-2 & -1 & 1\\ 2 & 2 & 1\\ 2 & 1 & 0\end{bmatrix}$
You should express the results of your four transformations in terms of the ordered basis $\mathscr C,$ not $\mathscr B$ (it appears that you did that to get a partially-correct answer, but your attempt suggests some incomplete understanding): $$\begin{align} T(1,1,1,1) & = 2 + 2x + 2x^2\\ & = -2(1) + 2(1+x) + 2(1+x^2)\\\\ T(1,1,1,0) & = 2 + 2x + x^2\\ & = -1(1) + 2(1+x) + 1(1+x^2)\\\\ T(1,1,0,0) & = 2 + x\\ & = 1(1) + 1(1+x) + 0(1+x^2)\\\\ T(1,0,0,0) & = 1\\ & = 1(1) + 0(1+x) + 0(1+x^2).\end{align}$$ The coefficients of the elements of $\mathscr C$ are the columns of the matrix of $T$ relative to $\mathscr B$ and $\mathscr C:$ $$\begin{bmatrix} -2 & -1 & 1 & 1\\ 2 & 2 & 1 & 0\\ 2 & 1 & 0 & 0\end{bmatrix}.$$