Solutions to Diophantine Equation x (x + 1) = y (y + 5) (y + 10) (y + 15)

Theorem
The Diophantine equation
 * $n = x \left({x + 1}\right) = y \left({y + 5}\right) \left({y + 10}\right) \left({y + 15}\right)$

has exactly $2$ solutions: