For what value of $x$ is the following number a perfect square

Question

I've the following number:

$$1+12x^2(1+x)$$

For what value of $x\ge2$ and $x\in\mathbb{N}$ is the number $1+12x^2(1+x)$ a perfect square?

Answer 1

$y^2=12x^2(1+x)+1\tag{1}$
Multiplying $12$ on both sides and let $X=12x, Y=12y$ then we get

$Y^2 = X^3+12X^2+144\tag{2}$

Using online Magma calculator as follows.

$E$:=EllipticCurve($[0, 12, 0, 0, 144]$);
IntegralPoints($E$);

It says that all integral points are $ [ (-12 : 12 : 1), (-8 : -20 : 1), (-3 : 15 : 1), (0 : -12 : 1), (4 : 20 : 1), (12 : -60 : 1), (48 : -372 : 1), (72 : 660 : 1), (1717 : -71395 : 1) ]$.

Hence all positive integral points are $(x,y)=(4,31),(6,55)$ with $x\geqq2.$