# Solving diophantine equation in two variables

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?

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