Solution:

Option(C)
To find: Value of $\left|\begin{array}{ccc}a^{2}+2 a & 2 a+1 & 1 \\ 2 a+1 & a+2 & 1 \\ 3 & 3 & 1\end{array}\right|$
We have, $\left|\begin{array}{ccc}a^{2}+2 a & 2 a+1 & 1 \\ 2 a+1 & a+2 & 1 \\ 3 & 3 & 1\end{array}\right|$
Applying $\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}$
$\Rightarrow\left|\begin{array}{ccc}
a^{2}-1 & a-1 & 0 \\
2 a+1 & a+2 & 1 \\
3 & 3 & 1
\end{array}\right|$
Applying $\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}$
$\Rightarrow\left|\begin{array}{ccc}
a^{2}-1 & a-1 & 0 \\
2 a-2 & a-1 & 0 \\
3 & 3 & 1
\end{array}\right|$
Expanding along $C_{3}$
$\begin{array}{l}
\Rightarrow\left[1\left\{\left(a^{2}-1\right)(a-1)-(a-1)(2 a-2)\right\}\right] \\
\Rightarrow[1\{(a-1)(a+1)(a-1)-(a-1) 2(a-1)\}] \\
\Rightarrow\left[\left\{(a+1)(a-1)^{2}-2(a-1)^{2}\right\}\right] \\
\Rightarrow\left[\left\{(a-1)^{2}(a+1-2)\right\}\right] \\
\Rightarrow\left[\left\{(a-1)^{2}(a-1)\right\}\right] \\
\Rightarrow(a-1)^{3}
\end{array}$