Example 3 Given the real-coefficient equation $x^{3}+2(k-1) x^{2}+9 x+5(k-1)=0$ has a complex root with a modulus of $\sqrt{5}$, find the value of $k$, and solve this equation.



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Example 3 Given the real-coefficient equation $x^{3}+2(k-1) x^{2}+9 x+5(k-1)=0$ has a complex root with a modulus of $\sqrt{5}$, find the value of $k$, and solve this equation.