Change the abnormal variable names to $a = dx$ and $b = dy$ so that the equation becomes $d+dxy = 4d(x+y) + 2021$, this means $d | 2021$ and so $d = \{1,43,47,2021\}$ Rewrite the equation as $d(x-4)(y-4) = 2021 + 15d$. So $d = 1$ means $(x-4)(y-4) = 2036$. Checking, see that the solutions to this are $(x,y) = (8,513), (513,8)$ since $x,y$ are coprime $d = 2021$ means that $(x-4)(y-4) = 16$ which has no solutions if $x,y$ are coprime. $d = 43$ gives that $(x-4)(y-4) = 47+15 = 62$, which has solutions $(x,y) = (5,66), (66,5), (6,35), (35,5)$ $d = 47$ gives $(x-4)(y-4) = 58$ which has solutions $(x,y) = (5,63), (63,5)$. So, finally, we see that all the pairs satisfying the given equation are : $(\alpha, \beta) = (8,513), (513,8), (2838, 215), (215, 2838), (1505, 258), (258, 1505), (235, 2961), (2961, 235)$