Formula for calculating the total number of solution pairs to an equation

Question

I was solving a mathematical olympiad problem, which reads as follows:

How many positive integer solution pair $(x, y)$ are there for the equation $y^2=\frac{x^5-1}{x-1}$, where $x\ne1$?

My first act was to simplify the equation to this:

$$y^2=x^4+x^3+x^2+x+1$$

Seeing that this did no good, I hit on the idea that this problem was actually very simple. A good, little trick would do it.

I decided to work with the first five integers (excluding $1$, of course). By trial and error, I discovered that there was only one integer solution pair i.e. $(3, 11)$ that satisfies the equation. This was as I suspected before. I also build a C program to confirm this; I was right.

But...

if I were to solve another such problem that doesn't involve tricks, then what? There should be a formula for calculating total number of solution pairs to equations of that kind. Is there? Could I solve this very problem using a formula?

Please answer bearing in mind that I'm only 12 years and a 7th grader.

How my question isn't a duplicate of any other question?

At first look, my question might seem to be the exact duplicate of this: $y^2 = \frac{x^5 - 1}{x-1}$ & $x,y \in \mathbb{Z}$. However, it isn't.

First of all, it's obvious that I didn't notice that question before posting this question.

The main difference lies in the question content and purpose. The other question was solely based on how to solve this particular problem. But, my question isn't that. I have solved the problem. My question is:

is there a formula in general to find out the number of rational solutions of a diophantine equation?

My question has a much more wider purpose. These reasons should be enough to convince you that this question isn't a duplicate of the question at all.

You could treat this as a degree four polynomial in $x$ and use the known formula to find all its roots (some of which may not be integers, and in fact not even real). — Fimpellizzeri, Dec 24 '16 at 16:14
See this link for greater detail. That said, on second thought, I think there probably should be a better way. — Fimpellizzeri, Dec 24 '16 at 16:18
This answer can be helpful: http://math.stackexchange.com/a/2066430/133781 — Xam, Dec 24 '16 at 16:32
Short answer to your ultimate question: no, there's no general universal formula for the number of solutions of all such equations. There are general enough methods for some types of Diophantine equations. But many such equations are one-of-a-kind, and that's why they require special tricks. — zipirovich, Dec 24 '16 at 18:11
This may be of help: https://math.stackexchange.com/questions/1835990/an-equation-in-mathbbz — Simply Beautiful Art, Dec 24 '16 at 18:20
@zipirovich Thank you for the short but definitive answer to my question. — Soha Farhin Pine, Dec 24 '16 at 18:24
@all Thanks. I've considered all the helpful ways of solving it. I'll definitely keep them in mind. — Soha Farhin Pine, Dec 24 '16 at 18:30
I do Discord. And if you are interested, I can make MathJax work on Discord, assuming you use Discord in a browser — Simply Beautiful Art, Dec 24 '16 at 18:34
See also: Determining all the positive integers $n$ such that $n^4+n^3+n^2+n+1$ is a perfect square. — Martin Sleziak, Jan 22 '18 at 11:58

Formula for calculating the total number of solution pairs to an equation

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