I feel a bit of a fraud coming in on this one after the introductory work done by the three of you, providing the ideas and so on, however 
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We need
$$\displaystyle \frac{x}{x^{2}+x+1}\quad \text{to equal} \quad 2.$$
Turn that upside down, as suggested by Chris, and we have
$$\displaystyle x+1+\frac{1}{x}=\frac{1}{2}\quad \dots(1).$$
Now look at the rhs, suppose that's equal to $$k$$.
Turn that upside down and we have
$$\displaystyle x^{2}+1+\frac{1}{x^{2}}=\frac{1}{k}.$$
Squaring (1),
$$\displaystyle x^{2}+1+\frac{1}{x^{2}}+2x+2+\frac{2}{x}=\frac{1}{4},$$
so
$$\displaystyle x^{2}+1+\frac{1}{x^{2}}= \frac{1}{4}-2\left(x+1+\frac{1}{x}\right)=\frac{1}{4}-\left(2\times\frac{1}{2}\right)=-\frac{3}{4},$$
in which case
$$\displaystyle \frac{1}{k}=-\frac{3}{4},\quad \text{so} \qquad k=-\frac{4}{3}.$$
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