The value of $\left|\begin{array}{ccc}(a+1)(a+2)& a+2& 1\\ (a+2)(a+3)& a+3& 1\\ (a+3)(a+4)& a+4& 1\end{array}\right|$ is :

The correct option is A $-2$
$\Delta =\left|\begin{array}{ccc}(a+1)(a+2)& a+2& 1\\ (a+2)(a+3)& a+3& 1\\ (a+3)(a+4)& a+4& 1\end{array}\right|$

Applying $C_1\to C_1-C_2,$
$\Delta =\left|\begin{array}{ccc}(a+2)a& a+2& 1\\ (a+3)(a+1)& a+3& 1\\ (a+4)(a+2)& a+4& 1\end{array}\right|$

Applying $C_2\to C_2-C_3,$
$\Delta =\left|\begin{array}{ccc}(a+2)a& a+1& 1\\ (a+3)(a+1)& a+2& 1\\ (a+4)(a+2)& a+3& 1\end{array}\right|$

Applying $R_2\to R_2-R_1$ and $R_3\to R_3-R_1,$
$\Delta =\left|\begin{array}{ccc}{a}^2+2a& a+1& 1\\ 2a+3& 1& 0\\ 4a+8& 2& 0\end{array}\right|=6-8=-2$