How do we see that the shape of an equation is closed? Actually, I would like to understand :
When we have an equation , can we decide that its shape closed curve? 
For instance, for the equation:
$$((x+y)^2) + 2((x^4)/4-(x^5)/5)=C$$
For which values of C makes the shape of this equation closed? 

For any value of $C$ the graph is closed. To see this, define a function $f:\Bbb R^2\to \Bbb R$ by $f(x,y)=(x+y)^2+2\left(\dfrac{x^4}{4}-\dfrac{x^5}{5}\right)-C$. Clearly $f$ is continuous and so $f^{-1}(\{0\})=\left\{(x,y)\in\Bbb R^2:(x+y)^2+2\left(\dfrac{x^4}{4}-\dfrac{x^5}{5}\right)=C\right\}$ is a closed set in $\Bbb R^2$ as this set is the inverse image of the closed subset $\{0\}$ under the continuous function $f$.
