We write it in vertex form

f(x) = a(x-h)² + k

f(x) = -1(x+2)² + 3
So -h = +2, so h = -2
and k = 3.

The vertex is the point (h,k) = (-2,3) which is a maximum POINT
if a is negative and a minimum POINT if a is positive.  Since
a = -1, then (-2,3) is a maximum POINT.  But you wanted the maximum
VALUE, not the maximum POINT.  The maximum VALUE is just the
y-coordinate of the maximum POINT, which is 3.

Here is the graph. Notice that it just reaches up as high as 3
on the y-axis.
 


Edwin