<s>[INST] Find the set of points on the complex plane for which $z^2 + z + 1$ is real and positive 
Find the set of points on the complex plane for which $z^2 + z + 1$ is  real and positive.
For being real I applied the condition: $z^2 + z + 1=\bar{z}^2 +
 \bar{z} + 1$
For the expression to be positive I applied:
$z^2+z+1>0 \implies (z+\dfrac{1+\sqrt{3}i}{2})(z+\dfrac{1-\sqrt{3}i}{2})>0 \implies z \in (-\infty,\dfrac{-1-\sqrt{3}i}{2}] \cup [\dfrac{-1+\sqrt{3}i}{2},\infty)$

But my doubt is it won't be correct to say $z$ is larger or smaller than a particular complex number as that makes no sense. What will be the correct method of solving this problem? 
 [/INST] $$z^2+z+1=k,\ k>0$$
$$z^2+z+(1-k)=0$$
$$z=\frac{-1\pm\sqrt{1^2-4\cdot1\cdot(1-k)}}{2\cdot1}$$
$$z=\frac{-1\pm\sqrt{4k-3}}2\tag1$$
When $0<k<\frac34$, $\sqrt{4k-3}$ is imaginary and $z=-\frac12\pm\frac{\sqrt{3-4k}}2i$; the range of the imaginary part is $(-\frac{\sqrt3}2,\frac{\sqrt3}2)$. When $k\ge\frac34$, $\sqrt{4k-3}$ is real and $z$ is given by $(1)$; its range is the whole real line.
In conclusion, the range of $z$ for which $z^2+z+1$ is real and positive is
$$\Bbb R\cup\{-\tfrac12+ai\mid a\in(-\tfrac{\sqrt3}2,\tfrac{\sqrt3}2)\}$$
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