Find the sum of the maximum and minimum

Question

For a real number $x$ find the sum of the maximum and minimum. $$y=\frac{x^2-2x-3}{2x^2+2x+1}$$ This is a sample question for a math competition. It seems like calculus would be used to solve this, but that sort of material shouldn't be on this competition.

Answer 1

$(2y-1)x^2+(2y+2)x+y+3=0 , \Delta=(2y+2)^2-4(2y-1)(y+3) \ge 0 \implies$

$4y^2+12y-16 \le 0 \iff (y+4)(y-1) \le0 \iff -4 \le y \le 1$