$\begin{array}{l} 3{x^2} - {y^2} - 2xy - 2x - 2y + 40 = 0\\ \Leftrightarrow 4{x^2} - \left( {{x^2} + {y^2} + 1 + 2xy + 2x + 2y} \right) + 41 = 0\\ \Leftrightarrow {\left( {2x} \right)^2} - {\left( {x + y + 1} \right)^2} = - 41\\ \Leftrightarrow {\left( {x + y + 1} \right)^2} - {\left( {2x} \right)^2} = 41\\ \Leftrightarrow \left( {3x + y + 1} \right)\left( { - x + y + 1} \right) = 41\\ \Leftrightarrow \left[ \begin{array}{l} \left\{ \begin{array}{l} 3x + y + 1 = 41\\ - x + y + 1 = 1 \end{array} \right.\\ \left\{ \begin{array}{l} 3x + y + 1 = - 41\\ - x + y + 1 = - 1 \end{array} \right.\\ \left\{ \begin{array}{l} 3x + y + 1 = 1\\ - x + y + 1 = 41 \end{array} \right.\\ \left\{ \begin{array}{l} 3x + y + 1 = - 1\\ - x + y + 1 = - 41 \end{array} \right. \end{array} \right. \Leftrightarrow \left[ \begin{array}{l} x = y = 10\\ \left\{ \begin{array}{l} x = - 10\\ y = - 12 \end{array} \right.\\ \left\{ \begin{array}{l} x = - 10\\ y = 30 \end{array} \right.\\ \left\{ \begin{array}{l} x = 10\\ y = - 32 \end{array} \right. \end{array} \right. \end{array}$
