jmc

We will complete the square on the given quadratic expression to find the vertex. Factoring 2 from the first two terms, we have $$y=2(x^2-2x)+4$$We complete the square inside the parentheses by adding $+1-1$ inside the parentheses to get $$y=2(x^2-2x+1-1)+4=2(x-1{)}^2+2$$The graph of an equation of the form $y=a(x-h{)}^2+k$ is a parabola with vertex at $(h,k)$, so the vertex of our parabola is at $(1,2)$. Thus, $m=\boxed{1}$.