Y = 2x square - 4x + 3 / x square + 2x + 3 to find the range of this equation? | Math enthusiast | Mathematical art

Y = 2x square - 4x + 3 / x square + 2x + 3 to find the range of this equation?

y=(2x^2-4x+3)/(x^2+2x+3)
2x^2-4x+3=yx^2+2yx+3y
(2-y)x^2-(4+2y)x+3(1-y)=0
Because the equation about X has a solution, the discriminant of the root is greater than or equal to zero,
(4+2y)^2-12(y-1)(y-2)≥0
4y^2+16y+16-(12y^-36y+24)≥0
-8y^2+52y-8≥0
2y^2-13y+2≥0
Y ≥ (13 + √ 153) / 4, or Y ≤ (13 - √ 153) / 4

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