Solving diophantine equation in two variables

Question

We need to find all positive integer solutions for the equation: $$ {x^2+6 x y+ 10 x+30 y -1470}= 0$$ How can we determine these solutions?

Answer 1

If $ x = u - 3 y-5$, the equation becomes $ u^2 - 9 y^2 = 1495$, i.e. $(u-3y)(u+3y) = 1495$. So consider all factorizations of $1495$...

Answer 2

The above expression factors to: $$ (x+5)(x+5+6y) -25 -1470 = 0$$ $$ (x+5)(x+5+6y) =1495$$

prime factors of 1495 are 5,13,23

pairwise these are (5,299), (13,115) and (23,65) (and their negative counterparts) which are all suitable for generating solutions.