Without expanding show that $\begin{vmatrix} x^2+x & x+1 & x-2 \\ 2x^2 +3x-1 & 3x & 3x-3 \\ x^2+2x+3 & 2x-1 & 2x-1 \end{vmatrix} = xA +B$ Also find A and B.

Question

$(A)\; A = \begin{vmatrix} 1 & 1 & 1 \\ -4 & 0 & 0 \\ 3 & -3 & 3 \end{vmatrix} \;and \; B= \begin{vmatrix} 0 & 1 & -2\\ -4 & 0 & 0 \\ 3 & -3 & 3 \end{vmatrix}$
$(B)\; A = \begin{vmatrix} 1 & -1 & 1 \\ -4 & 0 & 0 \\ 3 & -3 & 3 \end{vmatrix} \;and \; B= \begin{vmatrix} 0 & 1 & 2\\ -4 & 0 & 0 \\ 3 & -3 & 3 \end{vmatrix}$
$(C)\; A = \begin{vmatrix} 1 & 1 & -1 \\ -4 & 0 & 0 \\ 3 & -3 & 3 \end{vmatrix} \;and \; B= \begin{vmatrix} 0 & -1 & -2\\ -4 & 0 & 0 \\ 3 & -3 & 3 \end{vmatrix}$
$(D)\; A = \begin{vmatrix} 1 & 1 & 1 \\ -4 & 0 & 0 \\ -3 & 3 & 3 \end{vmatrix} \;and \; B= \begin{vmatrix} 0 & 1 & -2\\ -4 & 0 & 0 \\ -3 & 3 & 3 \end{vmatrix}$