This is a downward facing parabola, so its range comes in the form
[tex](-\infty, m)[/tex] where m is the y coordinate of the vertex of the parabola.
We can find the x coordinate of the vertex using
[tex]x=-\dfrac{b}{2a}=-\dfrac{-2}{-2}=-1[/tex]
And if we plug -1 in the equation we get the y coordinate of the vertex:
[tex]y=-(-1)^2-2(-1)+3 = -1+2+3=4[/tex]
So, the range is the interval
[tex](-\infty, 4)[/tex]
