Explanation

(d)
We have, $\left| \begin{matrix}
x^{2} + x & x + 1 & x - 2 \\
2x^{2} + 3x - 1 & 3x & 3x - 3 \\
x^{2} + 2x + 3 & 2x - 1 & 2x - 1
\end{matrix} \right| = Ax - 12$
On putting $x = 1$ on both sides, we get
$\left| \begin{matrix}
2 & 2 & - 1 \\
4 & 3 & 0 \\
6 & 1 & 1
\end{matrix} \right| = A - 12$
Applying $C_{1} \rightarrow C_{1} - C_{2}$
$\Rightarrow \ \left| \begin{matrix}
0 & 2 & - 1 \\
1 & 3 & 0 \\
5 & 1 & 1
\end{matrix} \right| = A - 12$
$\Rightarrow \ - 2(1) + ( - 1)( - 14) = A - 12$
$\Rightarrow A = 24$