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(A) Let $f(x) = \left|\begin{array}{ccc}x^2+3x & x+1 & x-3 \ x-1 & 2-x & x+4 \ x-3 & x-3 & 3x\end{array}ight| = a_0+a_1x+a_2x^2+a_3x^3+a_4x^4$. Pu

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(A) Let $f(x) = \left|\begin{array}{ccc}x^2+3x & x+1 & x-3 \ x-1 & 2-x & x+4 \ x-3 & x-3 & 3x\end{array}ight| = a_0+a_1x+a_2x^2+a_3x^3+a_4x^4$. Put $x=1$ in the equation: $f(1) = \left|\begin{array}{ccc}4 & 2 & -2 \ 0 & 1 & 5 \ -2 & -2 & 3\end{array}ight| = a_0+a_1+a_2+a_3+a_4$. Calculating the determinant: $4(3+10) - 2(0+10) - 2(0+2) = 4(13) - 2(10) - 2(2) = 52 - 20 - 4 = 28$. So, $a_0+a_1+a_2+a_3+a_4 = 28$ (Eq. $i$). Put $x=-1$ in the equation: $f(-1) = \left|\begin{array}{ccc}-2 & 0 & -4 \ -2 & 3 & 3 \ -4 & -4 & -3\end{array}ight| = a_0-a_1+a_2-a_3+a_4$. Calculating the determinant: $-2(-9+12) - 0 + (-4)(8+12) = -2(3) - 4(20) = -6 - 80 = -86$. So, $a_0-a_1+a_2-a_3+a_4 = -86$ (Eq. $ii$). Subtracting Eq. $ii$ from Eq. $i$: $(a_0+a_1+a_2+a_3+a_4) - (a_0-a_1+a_2-a_3+a_4) = 28 - (-86) = 114$. $2(a_1+a_3) = 114 \Rightarrow a_1+a_3 = 57$. Adding Eq. $i$ and Eq. $ii$: $(a_0+a_1+a_2+a_3+a_4) + (a_0-a_1+a_2-a_3+a_4) = 28 + (-86) = -58$. $2(a_0+a_2+a_4) = -58$. Therefore, $(a_1+a_3) + 2(a_0+a_2+a_4) = 57 + (-58) = -1$.

If $\left|\begin{array}{ccc}x^2+3x & x+1 & x-3 \\ x-1 & 2-x & x+4 \\ x-3 & x-3 & 3x\end{array}\right|=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4$, then find the value of $(a_1+a_3)+2(a_0+a_2+a_4)$.

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