Find the number of integral solutions of $(x,y)$ Given this equation: $4x^3+5=y^2$

Find the ordered pairs of $(x,y)$ where $x,y\in Z$
This is an elliptic curve, and it would appear that it has infinitely-many rational points (generated by (1,3)).  It is also an example of "Mordell's Equation" - curves of the form $y^2 = x^3 + D$ (in your case D = 80).  Many things are known about its integral solutions.  You might find this article by Keith Conrad to be interesting.  The Wikipedia article on the subject links to a large source of data, as well.
$4x^3+5=y^2$

(1)$4(x^3+1)=y^2-1$ Therefore $x=-1,y=∓1$ are solutions

(2) $4(x^3-1)=y^2-9$ Therefore $x=+1,y=∓3$ are solutions

You need integer solutions, No?. Then there is  no any other integer solutions. If you can’t prove this using High school mathematics  let me know.

P.Ranawaka