When is the number $81 + 60 x (1 + x) (-2 + 5 x)$ a perfect square for $x\ge2$ and $x\in\mathbb{N}$

Question

I've the following number:

$$81 + 60 x (1 + x) (-2 + 5 x)$$

For what value of $x\ge2$ and $x\in\mathbb{N}$ is the number $81 + 60 x (1 + x) (-2 + 5 x)$ a perfect square?

Answer 1

We have the equation $y^2 = 300 x^3 + 180 x^2 - 120 x + 81$.

We re-write it as $(300y)^2 = (300x)^3 + 180(300x)^2 - 36000(300x) + 7290000$.

Now paste the follow codes

E = EllipticCurve([0, 180, 0, -36000, 7290000])
P = E.integral_points()
for p in P:
    if p[0] % 300 == 0:
        print(p[0] // 300, p[1] // 300)

in this page and press "Evaluate", and we get all integral solutions: $(x, y) = (-1, 9), (0, 9), (1, 21)$.