Show that 1/3 ≤ (x^{2}-2x+4)/(x^{2}+2x+4) ≤ 3 for all real values of x.

1/3 ≤ (x^{2}-2x+4)/(x^{2}+2x+4) ≤ 3

let y=(x^{2}-2x+4)/(x^{2}+2x+4)

yx^{2}+2yx+4y=x^{2}-2x+4

(y-1)x^{2}+(2y+2)x+4y-4=0

if x is real, then b^{2}-4ac ≥ 0

(2y-2)^{2}-4(y-1)(4y-4)≥0

y^{2}-2y+1≤0

(y-1)^{2}≤0

any idea???