Why does $3x^2+y^3+9=-2xy$ cannot be written in explicit form?

Question

For this equation

$$3x^2 + y^3 + 9 = -2xy,$$

I checked on wolfram alpha and it doesn't seem to have an explicit form where $y$ is isolated and equals a function of $x$'s only.

The problem is that I don't understand why it can't be written explicitly, as the graph of the relation is a [non injective] function : it doesn't have more than one value of $y$ for each value of $x$.

So I should be able to write it as any normal function where $y = f(x)$ and its derivative with respect to $x$ should be expressed with only $x$'s instead of $x$'s and $y$'s - its derivative is : $\frac{-2(3x+y)}{3y^2+2x}$.

Please illuminate me, i'm really clueless.

Answer 1

According to wolfram alpha, there does exist a real explicit form: