Find all possible positive integers $x$ and $y$ such that the equation: $(x+y)(x-y)=\frac{(y+1)(y-1)}{24}$ is satisfied.

The equation is equivalent to $$ 24x^2 - 25y^2 =- 1. $$ An equation of the form $ax^2-by^2=c$ can be solved by continued fractions, see here, or by the LMM method. The solutions are given by the family $$ x = u + 25 v, y = u + 24 v $$ with $u^2-600v^2=1$. This is Pell's equation. Its fundamental solution is $(u,v)=(49,2)$, which gives $(x,y)=(99,97)$. So we have infinitely many solutions.