a.  \(x^2\left(y-1\right)+y^2\left(x-1\right)=1\)

<=> \(x^2y+y^2x-\left(x^2+y^2\right)=1\)

<=> \(xy\left(x+y\right)-\left(x+y\right)^2+2xy=1\)

Đặt: x + y = u; xy = v => u; v là số nguyên

Ta có: uv - \(u^2+2v=1\)

<=> \(u^2-uv-2v+1=0\) 

<=> \(u^2+1=v\left(2+u\right)\)

=> \(u^2+1⋮2+u\)

=> \(u^2-4+5⋮2+u\)

=> \(5⋮2-u\)

=> 2 - u = 5; 2 - u = -5; 2- u = 1; 2- u = -1 

Mỗi trường hợp sẽ tìm đc v 

=> x; y 

\(\Leftrightarrow\sqrt{9x^2+16x+96}=3x-16y-24\)

Vế phải nguyên \(\Rightarrow\) vế trái nguyên

\(\Rightarrow9x^2+16x+96=k^2\)

\(\Rightarrow81x^2+144x+864=\left(3k\right)^2\)

\(\Leftrightarrow\left(9x+8\right)^2+800=\left(3k\right)^2\)

\(\Leftrightarrow\left(3k-9x-8\right)\left(3k+9x+8\right)=800\)

Pt ước số thật kinh dị với số ước của 800 

Ta có \(9x^2+16x+96=\left(3x-24-16y\right)^2\)

\(\Leftrightarrow9x^2+16x+96=9x^2-6x\left(16y+24\right)+\left(16y+24\right)^2\)\(\Leftrightarrow16x+96=\left(16y+24\right)\left(16y+24-6x\right)\)

\(\Leftrightarrow8\left(2x+12\right)=4\left(4y+6\right).2\left(8y+12-3x\right)\)

\(\Leftrightarrow2x+12=\left(4y+6\right)\left(8y+12-3x\right)\)\(\Leftrightarrow2x+12=32y^2+48y-12xy+48y+72-18x\)

\(\Leftrightarrow32y^2+96y-12xy-20x+60=0\)\(\Leftrightarrow32y^2+96y+60=12xy+20x\)\(\Leftrightarrow8y^2+24y+15=3xy+5x\)

\(\Leftrightarrow8y^2+24y+15=x\left(3y+5\right)\)\(\Leftrightarrow x=\dfrac{8y^2+24y+15}{3y+5}\)

\(\Leftrightarrow9x=\dfrac{9\left(8y^2+24y+15\right)}{3y+5}=\dfrac{72y^2+216y+135}{3y+5}\)\(=\dfrac{\left(72y^2+120y\right)+\left(96y+160\right)-25}{3y+5}\)\(=24y+32-\dfrac{25}{3y+5}\)

\(\Leftrightarrow24y+32-\dfrac{25}{3y+5}\in Z\)\(\Rightarrow3y+5\in U\left(25\right)=\left\{\pm1,\pm5,\pm25\right\}\)\(\Leftrightarrow3y\in\left\{-4,-6,-10,0,-30,20\right\}\)\(\Rightarrow y\in\left\{-2,-10,0\right\}\)

+) Với y=-2=> x=1

+) với y=-10=> x=-23    

Vậy pt cho 2 cặp (x,y) nguyên =(1,-2),(-23,-10)

\(\Leftrightarrow9x^2-6x\left(16y+24\right)+\left(16y+24\right)^2=9x^2+16x+32\)

\(\Leftrightarrow x\left(3y+5\right)=8y^2+24y+17\)

\(\Leftrightarrow x=\dfrac{8y^2+24y+17}{3y+5}\in Z\)

\(\Rightarrow9x=\dfrac{9\left(8y^2+24y+17\right)}{3y+5}\in Z\)

\(\Rightarrow24y+62-\dfrac{157}{3y+5}\in Z\)

\(\Rightarrow3y+5=Ư\left(157\right)=\left\{-157;-1;1;157\right\}\)

\(\Rightarrow y=...\)

\(PT\Leftrightarrow9x^2+16x+96=9x^2+256y^2+576-96xy+768y-144x.\)

\(\Leftrightarrow256y^2-160x-96xy+768y+480=0\)

\(\Leftrightarrow8y^2-5x-3xy+24y+15=0\)

Đến chỗ này phân tích kiểu j được nhỉ

ảo quá

ĐKXĐ:\(9x^2+16x+32 ≥ 0 <=>(9x^2+12x+4)+4x+28≥0 <=>(3x+2)^2+4x+28 ≥0\)

 Mà \((3x+2)^2 ≥0\) 

\(=>4x+28 ≥0 =>x ≥-7\)

Phương trình\(<=> \)\((3x-16y-24)^2=9x^2+16x+32\)

Ta có:\(9x^2+16x+32=(3x+2)^2+4x+28 ≥(3x+2)^2\)