=>2[(x1+x2)^3-3x1x2(x1+x2)]+5x1x2(x1+x2)=-284

=>2[(-4)^3-3*(-4)(-2m-3)]+5(-2m-3)*(-4)=-284

=>2[-64+12(-2m-3)]+20(2m-3)=-284

=>-128+24(-2m-3)+40m-60=-284

=>40m-188-48m-72=-284

=>-8m-260=-284

=>8m=24

=>m=3

PT hoành độ giao điểm: \(2x^2=-2mx+m+1\)

\(\Leftrightarrow2x^2+2mx-\left(m+1\right)=0\)

Vì (P) cắt (d) tại 2 điểm phân biệt nên \(\Delta'=m^2+2\left(m+1\right)>0\)

\(\Leftrightarrow\left(m+1\right)^2>0\left(\text{đúng với mọi }m\ne-1\right)\)

Áp dụng Viét: \(\left\{{}\begin{matrix}x_1+x_2=-\dfrac{2m}{2}=-m\\x_1x_2=\dfrac{-\left(m+1\right)}{2}\end{matrix}\right.\)

Ta có \(\dfrac{1}{\left(2x_1-1\right)^2}+\dfrac{1}{\left(2x_2-1\right)^2}=2\)

\(\Leftrightarrow\dfrac{4x_2^2-4x_2+1+4x_1^2-4x_1+1}{\left[\left(2x_1-1\right)\left(2x_2-1\right)\right]^2}=2\\ \Leftrightarrow4\left[\left(x_1+x_2\right)^2-2x_1x_2\right]-4\left(x_1+x_2\right)+2=2\left[4x_1x_2-2\left(x_1+x_2\right)+1\right]^2\\ \Leftrightarrow4\left(m^2+m+1\right)+4m=2\left(-2m-2+2m+1\right)^2\\ \Leftrightarrow4m^2+4m+4+4m=2\\ \Leftrightarrow2m^2+4m+1=0\\ \Leftrightarrow\left[{}\begin{matrix}m=\dfrac{-2+\sqrt{2}}{2}\left(tm\right)\\m=\dfrac{-2-\sqrt{2}}{2}\left(tm\right)\end{matrix}\right.\)

Ta có: \(x_1^2x_2+x_2^2x_1-x_1x_2=3\)

\(\Leftrightarrow x_1x_2\left(x_1+x_2\right)-x_1x_2=3\)

\(\Leftrightarrow x_1x_2\cdot\left(x_1+x_2-1\right)=3\)

a: \(a_1+a_2=2x_1-x_2+2x_2-x_1=x_1+x_2=7\)

\(a_1a_2=\left(2x_1-x_2\right)\left(2x_2-x_1\right)\)

\(=4x_1x_2-2x_1^2-2x_2^2+x_1x_2\)

\(=5x_1x_2-2\left(x_1^2+x_2^2\right)\)

\(=5x_1x_2-2\left[\left(x_1+x_2\right)^2-2x_1x_2\right]\)

\(=5\cdot3-2\left[7^2-2\cdot3\right]\)

\(=15-2\left[49-6\right]\)

\(=15-2\cdot43=15-86=-71\)

Do đó: Pt cần tìm là \(a^2-7a-71=0\)

b: \(A^2=\left[\left(2x_1-x_2\right)^2+\left(2x_2-x_1\right)^2+2\left(2x_1-x_2\right)\left(2x_2-x_1\right)\right]\)
\(=\left[4x_1^2-4x_1x_2+x_2^2+4x_2^2-4x_2x_1+x_1^2+2\cdot\left(-71\right)\right]\)

\(=\left[5\left(x_1^2+x_2^2\right)-8x_1x_2+2\cdot\left(-71\right)\right]\)

\(=\left[5\cdot43-8\cdot3-142\right]\)

\(=49\)

=>A=7 hoặc A=-7 

Áp dụng Viét có: `{(x_1+x_2=-b/a=-9/2),(x_1.x_2=c/a=-3):}`

Ta có:`B=4(x_1 ^2+x_2 ^2)+5x_1.x_2`

`<=>B=4(x_1+x_2)^2-8x_1.x_2+5x_1.x_2`

`<=>B=4(-9/2)^2-3.(-3)`

`<=>B=90`

         `x^2 - 2 ( m + 2 ) x + m^2 + 7 = 0` `(1)`

`a)` Thay `m = 1` vào `(1)`. Ta có:

     `x^2 - 2 ( 1 + 2 ) x + 1^2 + 7 = 0`

`<=> x^2 - 6x + 8 = 0`

Ptr có: `\Delta' = b'^2 - ac = (-3)^2 - 8 = 1 > 0`

  `=>` Ptr có `2` `n_o` pb

`x_1 = [ -b' + \sqrt{\Delta'} ] / a = [ -(-3) + \sqrt{1} ] / 1 = 4`

`x_2 = [ -b' - \sqrt{\Delta'} ] / a = [ -(-3) - \sqrt{1} ] / 1 = 2`

Vậy với `m = 1` thì `S = { 2 ; 4 }`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`b)` Ptr `(1)` có nghiệm `<=> \Delta' >= 0`

                                     `<=> b'^2 - ac >= 0`

                                     `<=> [ - ( m + 2 ) ]^2 - ( m^2 + 7 ) >= 0`

                                     `<=> m^2 + 4m + 4 - m^2 - 7 >= 0`

                                     `<=> 4m - 3 >= 0`

                                     `<=> m >= 3 / 4`

Với `m >= 3 / 4`, áp dụng Vi-ét: `{(x_1 + x_2 = [-b] / a = 2m +4),(x_1 . x_2 = c / a = m^2 + 7):}`

Ta có: `-2x_1 + x_1 . x_2 - 2x_2 = 4`

  `<=>x_1 . x_2 - 2 ( x_1 + x_2 ) = 4`

  `<=> m^2 + 7 - 2 ( 2m +4 ) = 4`

  `<=>m^2 + 7 - 4m - 8 - 4 = 0`

  `<=> m^2 - 4m -5 = 0`

Ptr có: `\Delta' = b'^2 - ac = (-2)^2 - (-5) = 9 > 0`

`=>` Ptr có `2` `n_o` pb

`m_1 = [ -b' + \sqrt{\Delta'} ] / a = -(-2) + \sqrt{9} = 5`  (t/m)

`m_2 = [ -b' - \sqrt{\Delta'} ] / a = -(-2) - \sqrt{3} = -1` (ko t/m)

Vậy `m = 5` thì ptr có `2` nghiệm t/m yêu cầu đề bài

\(∘Angel\)

\(a)\) Thay \(m=1\) vào \((1)\) cta có : 

\(x^2− 2 ( 1 + 2 ) x + 1 ^2 + 7 = 0\)

\(x ^2 − 6 x + 8 = 0\)

Pt có : \(Δ ' = b ' ^2 − a c = ( − 3 ) ^2 − 8 = 1 > 0\)

Pt có 2 \(n\)_\(o\) pb

\(x1=\dfrac{b'+\sqrt{\text{Δ '}}}{a}=\dfrac{-\left(-3\right)+\sqrt{1}}{1}=4\)

\(x2=\dfrac{-b'-\sqrt{\text{Δ '}}}{a}=\dfrac{-\left(-3\right)-\sqrt{1}}{1}=2\)

\(m=1\) thì \(S=\)\(\left\{2;4\right\}\)

= 1/8 x (1+2+5+2)

=1/8 x 10

= 10/8 = 5/4

5/4