A numerical 4×4 determinant

Add columns. $C_1\leftarrow C_1+C_2+C_3+C_4$: each entry of column 1 becomes $10$. So
$$\Delta = 10\begin{vmatrix} 1 & 2 & 3 & 4 \\ 1 & 3 & 4 & 1 \\ 1 & 4 & 1 & 2 \\ 1 & 1 & 2 & 3 \end{vmatrix}.$$

Row operations. $L_2\leftarrow L_2-L_1$, $L_3\leftarrow L_3-L_1$, $L_4\leftarrow L_4-L_1$:
$$\Delta = 10\begin{vmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 1 & -3 \\ 0 & 2 & -2 & -2 \\ 0 & -1 & -1 & -1 \end{vmatrix}.$$

Expand along column 1.
$$\Delta = 10\begin{vmatrix} 1 & 1 & -3 \\ 2 & -2 & -2 \\ -1 & -1 & -1 \end{vmatrix}.$$

Compute the 3×3. $L_2\leftarrow L_2-2L_1$, $L_3\leftarrow L_3+L_1$:
$$= 10\begin{vmatrix} 1 & 1 & -3 \\ 0 & -4 & 4 \\ 0 & 0 & -4 \end{vmatrix} = 10\cdot1\cdot(-4)\cdot(-4)=160.$$

Conclusion. $\Delta = 160$.