if x be real, prove that the value of 11x^2+12x+6/x^2+4x+2 cannot lie between -5 and 3 - Maths - Complex Numbers and Quadratic Equations

Question

if x be real, prove that the value of 11x^2+12x+6/x^2+4x+2
cannot lie between -5 and 3

Priyanka Kedia · Accepted Answer

Let y=11x2+12x+6x2+4x+2⇒yx2+4xy+2y=11x2+12x+6⇒x2y-11+x4y-12+2y-6=0Since x be real,therefore discriminant will be non negative
D≥0⇒4y-122-4y-112y-6≥0⇒16y2+144-96y-42y2-28y+66≥0⇒16y2+144-96y-8y2+112y-264≥0⇒8y2+16y-120≥0⇒y2+2y-15≥0⇒y+5y-3≥0⇒y∈(-∞,-5]∪[3,∞)⇒y can not lie between -5 and 3