Nếu \(x< -3\) thì \(x^2+x+3< x^2\) và \(x^2+x+3>\left(x+1\right)^2\), vô lý.

 Nếu \(x>2\) thì \(x^2+x+3>x^2\) và \(x^2+x+3< \left(x+1\right)^2\), cũng vô lý.

 Do đó \(x\in\left\{-3;-2;-1;0;1;2\right\}\)

 Thử từng giá trị, ta thấy \(\left(x;y\right)\in\left\{\left(-3;3\right);\left(-3;-3\right)\right\}\) là các cặp số thỏa ycbt.

\(x^2-25=y\left(y+6\right)\)

\(\Leftrightarrow x^2-25=y^2+6y\)

\(\Leftrightarrow x^2-25-y^2-6y=0\)

\(\Leftrightarrow x^2-\left(y^2+6y+9\right)-16=0\)

\(\Leftrightarrow x^2-\left(y+3\right)^2=16\)

\(\Leftrightarrow\left(x+y+3\right)\left(x-y-3\right)=16\)

\(\Leftrightarrow\left(x+y+3\right);\left(x-y-3\right)\in\left\{-1;1;-2;2;-4;4;-8;8;-16;16\right\}\)

Ta giải các hệ phương trình sau :

1) \(\left\{{}\begin{matrix}x+y+3=-1\\x-y-3=-16\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x+y=-4\\x-y=-15\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}2x=-11\left(loại\right)\\x-y=-15\end{matrix}\right.\)

2) \(\left\{{}\begin{matrix}x+y+3=1\\x-y-3=16\end{matrix}\right.\)  \(\Leftrightarrow\left\{{}\begin{matrix}x+y=-2\\x-y=19\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}2x=17\left(loại\right)\\x-y=19\end{matrix}\right.\)

3) \(\left\{{}\begin{matrix}x+y+3=2\\x-y-3=8\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x+y=-1\\x-y=11\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}2x=10\\x-y=11\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=5\\y=-6\end{matrix}\right.\)

4) \(\left\{{}\begin{matrix}x+y+3=-2\\x-y-3=-8\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x+y=-5\\x-y=-5\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}2x=-10\\x-y=-5\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=-5\\y=0\end{matrix}\right.\)

5) \(\left\{{}\begin{matrix}x+y+3=-4\\x-y-3=-4\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x+y=-7\\x-y=-1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}2x=-6\\x-y=-1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=-3\\y=-2\end{matrix}\right.\)

6) \(\left\{{}\begin{matrix}x+y+3=4\\x-y-3=4\end{matrix}\right.\)  \(\Leftrightarrow\left\{{}\begin{matrix}x+y=1\\x-y=7\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}2x=8\\x-y=7\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=4\\y=-3\end{matrix}\right.\)

7) \(\left\{{}\begin{matrix}x+y+3=-8\\x-y-3=-2\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x+y=-11\\x-y=1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}2x=-10\\x-y=1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=-5\\y=-6\end{matrix}\right.\)

8) \(\left\{{}\begin{matrix}x+y+3=8\\x-y-3=2\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x+y=5\\x-y=5\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}2x=10\\x-y=5\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x=5\\y=0\end{matrix}\right.\)

9) \(\left\{{}\begin{matrix}x+y+3=-16\\x-y-3=-1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x+y=-19\\x-y=2\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}2x=-17\left(loại\right)\\x-y=2\end{matrix}\right.\)

10) \(\left\{{}\begin{matrix}x+y+3=16\\x-y-3=1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x+y=15\\x-y=4\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}2x=19\left(loại\right)\\x-y=4\end{matrix}\right.\)

Vậy \(\left(x;y\right)\in\left\{\left(5;-6\right);\left(-5;0\right);\left(-3;-2\right);\left(4;-3\right);\left(-5;-6\right);\left(5;0\right)\right\}\)

\(y^2+2xy-3x-2=0\)

\(\Leftrightarrow\left(y^2+2xy+x^2\right)-\left(x^2+3x+2\right)=0\)

\(\Leftrightarrow\left(x+y\right)^2=\left(x+1\right)\left(x+2\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}x+1=0\\x+2=0\end{matrix}\right.\)

Nếu \(x+1=0\) thì \(\left\{{}\begin{matrix}x=-1\\y=1\end{matrix}\right.\). 

Nếu \(x+2=0\) thì \(\left\{{}\begin{matrix}x=-2\\y=2\end{matrix}\right.\)

Thử lại, ta thấy thỏa mãn. Vậy ta tìm được các cặp số \(\left(x;y\right)\) thỏa mãn đề bài là \(\left(-1;1\right),\left(-2;2\right)\)

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