Solve the system of equations by determinants

Using the principle of minors to evaluate determinants, we have

$D= \begin{vmatrix} 1 & -2 & 1 & 3 \\ 1 & 1 & 3 & 2 \\ 1 & 1 & -2 & 1 \\ 3 & 4 & 4 & 2 \end{vmatrix}=1 \begin{vmatrix} 1 & 3 & 2 & 1 & 3 \\ 1 & -2 & 1 & 1 & -2 \\ 4 & 4 & 2 & 4 & 4 \end{vmatrix}-1 \begin{vmatrix} -2 & 1 & 3 & -2 & 1 \\ 1 & -2 & 1 & 1 & -2 \\ 4 & 4 & 2 & 4 & 4 \end{vmatrix}+1 \begin{vmatrix} -2 & 1 & 3 & -2 & 1 \\ 1 & 3 & 2 & 1 & 3 \\ 4 & 4 & 2 & 4 & 4 \end{vmatrix}-3 \begin{vmatrix} -2 & 1 & 3 & -2 & 1 \\ 1 & 3 & 2 & 1 & 3 \\ 1 & -2 & 1 & 1 & -2 \end{vmatrix}$

$D=1[-4+12+8-(-16+4+6)]-1[8+4+12-(-24-8+2)]+1[-12+8+12-(36-16+2)]-3[-6+2-6-(9+8+1)]$

$D=22-54-14+84=38$

$N_x= \begin{vmatrix} 7 & -2 & 1 & 3 \\ 6 & 1 & 3 & 2 \\ 9 & 1 & -2 & 1 \\ 16 & 4 & 4 & 2 \end{vmatrix}=7 \begin{vmatrix} 1 & 3 & 2 & 1 & 3 \\ 1 & -2 & 1 & 1 & -2 \\ 4 & 4 & 2 & 4 & 4 \end{vmatrix}-6 \begin{vmatrix} -2 & 1 & 3 & -2 & 1 \\ 1 & -2 & 1 & 1 & -2 \\ 4 & 4 & 2 & 4 & 4 \end{vmatrix}+9 \begin{vmatrix} -2 & 1 & 3 & -2 & 1 \\ 1 & 3 & 2 & 1 & 3 \\ 4 & 4 & 2 & 4 & 4 \end{vmatrix}-16 \begin{vmatrix} -2 & 1 & 3 & -2 & 1 \\ 1 & 3 & 2 & 1 & 3 \\ 1 & -2 & 1 & 1 & -2 \end{vmatrix}$

$N_x=7[-4+12+8-(-16+4+6)]-6[8+4+12-(-24-8+2)]+9[-12+8+12-(36-16+2)]-16[-6+2-6-(9+8+1)]$

$N_x=154-324-126+448=152$

$N_y= \begin{vmatrix} 1 & 7 & 1 & 3 \\ 1 & 6 & 3 & 2 \\ 1 & 9 & -2 & 1 \\ 3 & 16 & 4 & 2 \end{vmatrix}=1 \begin{vmatrix} 6 & 3 & 2 & 6 & 3 \\ 9 & -2 & 1 & 9 & -2 \\ 16 & 4 & 2 & 16 & 4 \end{vmatrix}-1 \begin{vmatrix} 7 & 1 & 3 & 7 & 1 \\ 9 & -2 & 1 & 9 & -2 \\ 16 & 4 & 2 & 16 & 4 \end{vmatrix}+1 \begin{vmatrix} 7 & 1 & 3 & 7 & 1 \\ 6 & 3 & 2 & 6 & 3 \\ 16 & 4 & 2 & 16 & 4 \end{vmatrix}-3 \begin{vmatrix} 7 & 1 & 3 & 7 & 1 \\ 6 & 3 & 2 & 6 & 3 \\ 9 & -2 & 1 & 9 & -2 \end{vmatrix}$

$N_y=1[-24+48+72-(-64+24+54)]-1[-28+16+108-(-96+28+18)]+1[42+32+72-(144+56+12)]-3[21+18-36-(81-28+6)]$

$N_y=82-146-66+168=38$

$N_z= \begin{vmatrix} 1 & -2 & 7 & 3 \\ 1 & 1 & 6 & 2 \\ 1 & 1 & 9 & 1 \\ 3 & 4 & 16 & 2 \end{vmatrix}=1 \begin{vmatrix} 1 & 6 & 2 & 1 & 6 \\ 1 & 9 & 1 & 1 & 9 \\ 4 & 16 & 2 & 4 & 16 \end{vmatrix}-1 \begin{vmatrix} -2 & 7 & 3 & -2 & 7 \\ 1 & 9 & 1 & 1 & 9 \\ 4 & 16 & 2 & 4 & 16 \end{vmatrix}+1 \begin{vmatrix} -2 & 7 & 3 & -2 & 7 \\ 1 & 6 & 2 & 1 & 6 \\ 4 & 16 & 2 & 4 & 16 \end{vmatrix}-3 \begin{vmatrix} -2 & 7 & 3 & -2 & 7 \\ 1 & 6 & 2 & 1 & 6 \\ 1 & 9 & 1 & 1 & 9 \end{vmatrix}$

$N_z=1[18+24+32-(72+16+12)]-1[-36+28+48-(108-32+14)]+1[-24+56+48-(72-64+14)]-3[-12+14+27-(18-36+7)]$

$N_z=-26+50+58-120=-38$

$N_w= \begin{vmatrix} 1 & -2 & 1 & 7 \\ 1 & 1 & 3 & 6 \\ 1 & 1 & -2 & 9 \\ 3 & 4 & 4 & 16 \end{vmatrix}=1 \begin{vmatrix} 1 & 3 & 6 & 1 & 3 \\ 1 & -2 & 9 & 1 & -2 \\ 4 & 4 & 16 & 4 & 4 \end{vmatrix}-1 \begin{vmatrix} -2 & 1 & 7 & -2 & 1 \\ 1 & -2 & 9 & 1 & -2 \\ 4 & 4 & 16 & 4 & 4 \end{vmatrix}+1 \begin{vmatrix} -2 & 1 & 7 & -2 & 1 \\ 1 & 3 & 9 & 1 & 3 \\ 4 & 4 & 16 & 4 & 4 \end{vmatrix}-3 \begin{vmatrix} -2 & 1 & 7 & -2 & 1 \\ 1 & 3 & 6 & 1 & 3 \\ 1 & -2 & 9 & 1 & -2 \end{vmatrix}$

$N_w=1[-32+108+24-(-48+36+48)]-1(64+36+28-(-56-72+16)]+1[-96+24+28-(84-48+16)]-3[-54+6-14-(21+24+9)]$

$N_w=64-240-96+348=76$

$x=\dfrac{N_x}{D}=\dfrac{152}{38}=4$

$y=\dfrac{N_y}{D}=\dfrac{38}{38}=1$

$z=\dfrac{N_z}{D}=\dfrac{-38}{38}=-1$

$w=\dfrac{N_w}{D}=\dfrac{76}{38}=2$

The desired answer is $(4+1-1+25)^2=\boxed{36}$

Notes:

$D$ = determinant of the coefficients of the variables

$N_x$ = determinants taken from $D$ replacing the coefficients of $x$ by their corresponding constant terms leaving all other terms unchanged

$N_y$ = determinants taken from $D$ replacing the coefficients of $y$ by their corresponding constant terms leaving all other terms unchanged

$N_z$ = determinants taken from $D$ replacing the coefficients of $z$ by their corresponding constant terms leaving all other terms unchanged

$N_w$ = determinants taken from $D$ replacing the coefficients of $w$ by their corresponding constant terms leaving all other terms unchanged