\(a^3-b^3+c^3=-3abc\)
\(\Leftrightarrow a^3-3a^2b+3ab^2-b^3+c^3+3a^2b-3ab^2+3abc=0\)
\(\Leftrightarrow\left(a-b\right)^3+c^3+3ab\left(a-b+c\right)=0\)
\(\Leftrightarrow\left(a-b+c\right)\left[\left(a-b\right)^2-\left(a-b\right)c+c^2\right]+3ab\left(a-b+c\right)=0\)
\(\Leftrightarrow\left(a-b+c\right)\left[a^2-2ab+b^2-ac+bc+c^2+3ab\right]=0\)
\(\Leftrightarrow\left(a-b+c\right)\left(a^2+b^2+c^2-ac+bc+ab\right)=0\)
\(\Rightarrow\left[{}\begin{matrix}a-b+c=0\\a^2+b^2+c^2-ac+bc+ac=0\end{matrix}\right.\)
Xét \(a-b+c=0\Rightarrow\left\{{}\begin{matrix}a=b-c\\b=a+c\\c=b-a\end{matrix}\right.\)
\(G=\left(1-\dfrac{a}{b}\right)\left(1-\dfrac{c}{b}\right)\left(1+\dfrac{c}{a}\right)=\dfrac{b-a}{b}.\dfrac{b-c}{b}.\dfrac{a+c}{a}=\dfrac{c}{b}.\dfrac{a}{b}.\dfrac{b}{a}=1\)
Xét \(a^2+b^2+c^2-ac+bc+ac=0\Leftrightarrow2a^2+2b^2+2c^2-2ac+2bc+2ac=0\)
\(\Leftrightarrow\left(a-c\right)^2+\left(b+c\right)^2+\left(a+b\right)^2=0\)
\(\Rightarrow\left\{{}\begin{matrix}a-c=0\\b+c=0\\a+b=0\end{matrix}\right.\)\(\Rightarrow a=-b=c\)
\(\Rightarrow G=\left(1-\dfrac{a}{b}\right)\left(1-\dfrac{c}{b}\right)\left(1+\dfrac{c}{a}\right)=\left(1+\dfrac{b}{b}\right)\left(1+\dfrac{b}{b}\right)\left(1+\dfrac{a}{a}\right)=2.2.2=8\)
