jmc

Rearranging the expression, we have $$2x^2+8x+3y^2-24y+62$$We first complete the square in $x$. Factoring a 2 from the first two terms of the expression, we get $$2(x^2+4x)+3y^2-24y+62$$In order for the expression inside the parenthesis to be a perfect square, we need to add and subtract $(4/2{)}^2=4$ inside the parenthesis. Doing this, we have $$2(x^2+4x+4-4)+3y^2-24y+62\Rightarrow 2(x+2{)}^2+3y^2-24y+54$$Now we complete the square in $y$. Factoring a 3 from the $y$ terms in the expression, we get $$2(x+2{)}^2+3(y^2-8y)+54$$In order for the expression inside the second parenthesis to be a perfect square, we need to add and subtract $(8/2{)}^2=16$ inside the parenthesis. Doing this, we have $$2(x+2{)}^2+3(y^2-8y+16-16)+54\Rightarrow 2(x+2{)}^2+3(y-4{)}^2+6$$Since the minimum value of $2(x+2{)}^2$ and $3(y-4{)}^2$ is $0$ (perfect squares can never be negative), the minimum value of the entire expression is $\boxed{6}$, and is achieved when $x=-2$ and $y=4$.