$3 x^2-5 y^2\+7 x\+4 y-9$

$\=3\left(x^2\+\frac{7}{3}x\right)-5\left(y^2-\frac{4}{5}y\right)-9$

$\=3\left(x^2\+2\frac{7}{2\cdot 3}x\+{\left(\frac{7}{6}\right)}^2-{\left(\frac{7}{6}\right)}^2\right)-5\left(y^2-\frac{4}{5}y\right)-9$

$\=3\left({\left(x\+\frac{7}{6}\right)}^2-{\left(\frac{7}{6}\right)}^2\right)-5\left(y^2-\frac{4}{5}y\right)-9$

$\=3{\left(x\+\frac{7}{6}\right)}^2-3{\left(\frac{7}{6}\right)}^2-5\left(y^2-\frac{4}{5}y\right)-9$

$\=3{\left(x\+\frac{7}{6}\right)}^2-5\left(y^2-\frac{4}{5}y\right)-9-3{\left(\frac{7}{6}\right)}^2$

$\=3{\left(x\+\frac{7}{6}\right)}^2-5\left(y^2-2\frac{4}{2\cdot 5}y\+{\left(\frac{4}{10}\right)}^2-{\left(\frac{4}{10}\right)}^2\right)-\frac{157}{12}$

$\=3{\left(x\+\frac{7}{6}\right)}^2-5\left({\left(y-\frac{2}{5}\right)}^2-{\left(\frac{2}{5}\right)}^2\right)-\frac{157}{12}$

$\=3{\left(x\+\frac{7}{6}\right)}^2-5{\left(y-\frac{2}{5}\right)}^2\+5{\left(\frac{2}{5}\right)}^2-\frac{157}{12}$

$\=3{\left(x\+\frac{7}{6}\right)}^2-5{\left(y-\frac{2}{5}\right)}^2\+\frac{4}{5}-\frac{157}{12}$

$\=3{\left(x\+\frac{7}{6}\right)}^2-5{\left(y-\frac{2}{5}\right)}^2-\frac{737}{60}$

$3 x^2-5 y^2\+7 x\+4 y-9$

$3x^2-5y^2\+7x\+4y-9$

$\stackrel{\text{complete square}}{\=}$

$-5{\left(y-\frac{2}{5}\right)}^2\+3x^2\+7x-\frac{41}{5}$

$\stackrel{\text{complete square}}{\=}$

$-5{\left(y-\frac{2}{5}\right)}^2\+3{\left(x\+\frac{7}{6}\right)}^2-\frac{737}{60}$

Apply the CompleteSquare command from the Student Precalculus package.