Let $\mathbb{R}$ be the set of real numbers,$ f : \mathbb{R} \to \mathbb{R}$ a map, and $E$ the equivalence relation on $ ℝ $ defined by $E = \{(x,y) \in \mathbb{R} \times \mathbb{R} \mid f(x) = f(y) \}.$
Describe the partition of $\Bbb{R}$ in the following case:
$f(x) = 2x^2+4x+8$ for all $x \in \mathbb{R}$.
I worked out that every $x$ is in fact equivalent to $-x-2$ by observation, but I would like to know if there is some algorithmic way to find the partition.