Answer
Local minimum is $y=-7$ when $x = -1$
Work Step by Step
The local minimum value of $g(x) = 2x^2 + 4x -5$ can be determined from its vertex
Vertex = $\frac {-b}{2a} = \frac{-4}{4} = {-1}$
At x=-1, g(-1) =-7
So the local minimum is $y=-7$ when $x= -1$