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Answer:$\left|\begin{array}{lll} a & h & g \\ h & b & f \\ g & f & c \end{array}\right|$$\left|\begin{array}{lll} a & h & g \\ h & b & f \\ g & f & c \end{array}\right|=a b c-a f^{2}-c h^{2}+2 f g h-b g^{2}$    $=abc-af^{2}-ch^{2}+2fgh-bg^{2}$$\left|\begin{array}{lll} a & h & g \\ h & b & f \\ g & f & c \end{array}\right|=a b c-a f^{2}-c h^{2}+2 f g h-b g^{2}$$\left|\begin{array}{lll} a & h & g \\ h & b & f \\ g & f & c \end{array}\right|=a b c-a f^{2}-c h^{2}+2 f g h-b g^{2}$$\left|\begin{array}{lll} a & h & g \\ h & b & f \\ g & f & c \end{array}\right|=a b c-a f^{2}-c h^{2}+2 f g h-b g^{2}$

Hint: We will expand it w.r.t $R_{1}$

Given:  $\left|\begin{array}{lll} a & h & g \\ h & b & f \\ g & f & c \end{array}\right|=a b c-a f^{2}-c h^{2}+2 f g h-b g^{2}$$\left|\begin{array}{lll} a & h & g \\ h & b & f \\ g & f & c \end{array}\right|$

Solution: $\left|\begin{array}{lll} a & h & g \\ h & b & f \\ g & f & c \end{array}\right|$

Expanding w.r.t $R_{1}$

\begin{aligned} &=a\left|\begin{array}{ll} b & f \\ f & c \end{array}\right|-h\left|\begin{array}{ll} h & f \\ g & c \end{array}\right|+g\left|\begin{array}{ll} h & b \\ g & f \end{array}\right| \\ &=a\left(b c-f^{2}\right)-h(h c-g f)+g(h f-b g) \\ &=a b c-a f^{2}-h^{2} c+f g h+f g h-b g^{2} \\ &=a b c-a f^{2}-c h^{2}+2 f g h-b g^{2} \end{aligned}

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