A nice Diophantine equation. | You should know the method. | How to evaluate?

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This video explains how to evaluate x+y, given that 3xy+x+y=44, x, y are integers.

How to find integer solutions
Find integers satisfying equations
Find all integer solutions
Find all integers satisfying equations
Solve Diophantine equations
Solving Diophantine equations
Evaluate algebraic expressions
How to evaluate algebraic expressions
Evaluating algebraic expressions
  1. Maths Enhancer's Class
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Thank you for explaining. I solved the additional problem. My answer is x^2+y^2 = 29, 34, 226, 485, 841, and 7569.
The calculation and the showing how to solve need a lot of explaining. So, I will write the main points only.
At first, we get (7x+1)(7y+3)=612 from the given equation. [[ 612=2×2×3×3×17 ]]
Next, we get (7x+1, 7y+3) = (1, 612), (36, 17), (204, 3), (-6, -102), (-34, -18), (-153, -4) Concerning other cases, x and y cannot be any integer.
Therefore, (x, y) = (0, 87), (5, 2), (29, 0), (-1, -15), (-5, -3), (-22, -1) Thus, x^2+y^2 = 7569, 29, 841, 226, 34, 485 (There are six solutions.)
I calculated without calculators and I had to check more than 30 cases for (7x+1, 7y+3).
So, I may miss something. If my calculation or solving way includes something strange, please inform me.

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