The value of the determinant $Δ=\begin{vmatrix}x+2&x+3&x+5\\x+4&x+6&x+9\\x+8 &x+11& x+15\end{vmatrix}$, is

-2

Applying $R_2 → R_2 - R_1$ and $R_3 → R_3 - R_2$, we get

$Δ=\begin{vmatrix}x+2&x+3&x+5\\2&3&4\\4 &5& 6\end{vmatrix}$

$⇒Δ=2\begin{vmatrix}x&x&x+1\\2&3&4\\1 &1& 1\end{vmatrix}$  [Applying $R_1→ R_1-R_2$ and $R_3 → R_3-R_2$]

$⇒Δ=2\begin{vmatrix}x&0&1\\2&1&1\\1 &0& 0\end{vmatrix}$  [Applying $C_2 →C_2-C_1$ and $C_3 →C_3-C_2$]

$⇒Δ=-2$  [Expanding along $R_3$]