$$\frac{x^2+x+1}{x-1} = 0$$
Answer
$$ \begin{matrix}x_1 = -\dfrac{ 1 }{ 2 }+\dfrac{\sqrt{ 3 }}{ 2 }i & x_2 = -\dfrac{ 1 }{ 2 }-\dfrac{\sqrt{ 3 }}{ 2 }i \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{x^2+x+1}{x-1} &= 0&& \text{multiply ALL terms by } \color{blue}{ x-1 }. \\[1 em](x-1)\frac{x^2+x+1}{x-1} &= (x-1)\cdot0&& \text{cancel out the denominators} \\[1 em]x^2+x+1 &= 0&& \\[1 em] \end{aligned} $$
$ x^{2}+x+1 = 0 $ is a quadratic equation.
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