Well, I have the following function:
$$\text{y}\left(x\right):=12\alpha^2\cdot x^3+12\alpha\cdot x^2+12\alpha\left(1-\alpha\right)\cdot x+\left(2-3\alpha\right)^2\tag1$$
Where $\alpha\in\mathbb{N}$.
Question: I want to find $x$ such that $\text{y}\left(x\right)$ can be written as a perfect square. I already know that when $x=3\alpha^2-2$, we can write:
$$\text{y}\left(x\right)=\left[2+3\alpha\left(6\alpha\left(\alpha^2-1\right)+1\right)\right]^2\tag2$$
But I want to find other solutions for $x$.