Determinants can be tiresome

Algebra Level 3

If \[ \left | \begin{array}{ccc} x^{2}-2x+3 & 7x+2 & x+4 \\ 2x+7 & x^{2}-x+2& 3x \\ 3 & 2x-1 & x^{2}-4x+7 \\ \end{array} \right | =ax^{6}+bx^{5}+cx^{4}+dx^{3}+ex^{2}+fx+g\] then the value of \(g\) is

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