It is given that the origin is translated from $\left(0,0\right)$ to $\left(-1,2\right)$ then we can say that $\left(x,y\right)\to \left(X,Y\right)$

We know, $x=X+h,y=Y+k$

$\Rightarrow x=X-1,y=Y+2$

$2x^2+y^2-3x+5y-8=0$

$2{\left(X-1\right)}^2+{\left(Y+2\right)}^2-3\left(X-1\right)+5\left(Y+2\right)-8=0$

$2\left(X^2+1-2X\right)+Y^2+4+4Y-3X+3+5Y+2=0$

$2X^2+2-4X+Y^2+4+4Y-3X+3+5Y+2=0$

$2X^2-7X+Y^2+9Y+11=0$

For obtaining the transformed equation, now replace $X\to x \& Y\to y$,

$2x^2+y^2-7x+9y+11=0$