f(x) = -x² - 2x + 2

We know it is a maximum simply because the coefficient
of x² is negative, for that means the the parabola opens 
downward and therefore has a maximum at the peak, which 
is the vertex.

The formula for the x-coordinate of the vertex of any
parabola which is the graph of 
    
       y = ax² + bx + c 

is



[Notice carefully here that although -1 is given as as choice, 
-1 is only the x-coordinate of the maximum point, it is NOT
the maximum VALUE.  That's the value of y when x is -1.]

To find the y-coordinate (the VALUE of the maximum),
we substitute -1 for x:

f(x) = -x² - 2x + 2
f(-1) = -(-1)² - 2(-1) + 2
f(-1) = -(1)+2+2
f(-1) = 3

So the maximum value is 3.

Now we graph as a check:



Edwin