BG-JBMO-TST 2017 Problem 2

Solve the following equation over the integers $25x^2y^2+10x^2y+25xy^2+x^2+30xy+2y^2+5x+7y+6= 0.$

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.