When the axes are translated to the point  (-2,3) then the transformed equation of the curve $2x^2+4xy+5y^2-4x-22y+7=0$ is

$a=2,h=2,b=5,g=-2,f=-11,c=7$

$\left(\frac{{\displaystyle hf-bg}}{{\displaystyle ab-h^2}},\frac{{\displaystyle gh-af}}{{\displaystyle ab-h^2}}\right)=(-2,3)$$=(x_1,y_1)$

Transformed equation is

$a{X}^2+2hXY+b{Y}^2+(g{x}_1+f{y}_1+c)=0$

$\Rightarrow 2X^2+4XY+5Y^2-2(-2)-11\left(3\right)+7=0$

$\therefore$$2X^2+4XY+5Y^2-22=0$