Cách `1`:

`H = (12x + 13)/(x^2 + 2x + 3)`

`<=> H(x^2 + 2x + 3) = 12x + 13`

`<=> Hx^2 + 2Hx + 3H - 12x - 13 = 0`

`<=> Hx^2 + 2(H - 6)x + 3H - 13 = 0`

`+) H = 0 => x = - 13/12`

`+) H ne 0`.Xét

`\Delta' = (H - 6)^2 - H(3H - 13) = (9 - 2H)(H + 4)`

Để pt có nghiệm `=> Delta' >= 0 <=> (9 - 2H)(H + 4) >= 0`

`<=> (2H - 9)(H + 4) <= 0 <=> - 4 <= H <= 9/2`

DBXR:

`H_(Mi\n) = - 4 <=> x = - 5/2`

`H_(Max) = 9/2 <=> x = 1/3`

Cách `2`:

Xét : `H + 4 = (12x + 13)/(x^2 + 2x + 3) + 4 = (12x + 13 + 4(x^2 + 2x + 3))/(x^2 + 2x + 3)`

`= (4x^2 + 20x + 25)/(x^2 + 2x + 3) = ((2x + 5)^2)/((x + 1)^2 + 2) >= 0`

`=> H >= - 4`

DBXR `<=> x = - 5/2`

Xét : `H - 9/2 = (12x + 13)/(x^2 + 2x + 3) - 9/2 = (2(12x + 13) - 9(x^2 + 2x + 3))/(2(x^2 + 2x + 3))`

`= - (9x^2 - 6x + 1)/(2(x^2 + 2x + 3)) = - ((3x - 1)^2)/(2[(x + 1)^2 + 2]) <= 0`

`=> H <= 9/2`

DBXR `<=> x = 1/3`