I have an equation in the following form:
$$6mn+m+n=x$$ $$m,n,x\in\Bbb Z; \qquad0 < m,n$$
If I were given a value for $x$, how would I go about finding solutions to this equality for $m$ and $n$ or determining that there are no solutions for $x$ in this form?
Example: $x = 15 \implies6(1)(2)+1+2=15 \implies(m=1,n=2),(m=2,n=1)$
Hint $\,\ $ In order to solve the equation $\rm\,\ 6ab+a+b\, =\, 15,\ $ multiply by $\,6,\,$ then add $\,1\,$ to obtain
$$\rm (6a+1)(6b+1)\, =\, 91$$
Clearly the same method works generally, i.e.
$$\rm a\,(axy+bx+cy)+bc\, =\, (ax+c)(ay+b) $$
Remark $\ $ Dario Alpern has a web page Quadratic two integer variable equation solver that will solve any binary quadratic Diophatine equation, using ideas that go back to Lagrange over 200 years ago. The web page has an option to configure it to provide step-by-step solutions. You may find the output instructive. You might also find of interest some more recent optimizations of Lagrange's old algorithm in this paper by H. C. Williams et al. $ $ A new look at an old equation.