<s>[INST] Find all ordered pairs such that $x + y^2 = 2$ and $y + x^2 = 2$ How can I find all ordered pairs $(x,y)$ such that $x + y^2 = 2$ and     $y + x^2 = 2$?
A solution/explanation would be greatly appreciated.
If it helps, the solutions given in the textbook are 
x=-2, y=-2; x=1, y=1; x= (1+root 5)/2, y=(1-root 5)/2;            x= (1-root 5)/2, y=(1+root 5)/2 
 [/INST] If we leave $y$ alone in the second equation, we have $y = 2-x^2$. Then putting it in the first equation, we get $x+(2-x^2) = 2 \implies x^4-4x^2+x+2 = 0$. Now, notice that $x=1$ satisfies this equation, therefore we can say that the polynomial $x^4-4x^2+x+2$ can be factorized as 
$$x^4-4x^2+x+2 = (x-1)(Ax^3+Bx^2+Cx+D)$$ 
Here, we can find $A$ and $D$ easily because there is only one term to consider and $A = 1$, $D = -2$. Notice that after that, we can conclude that $B-1 = 0$ and $C-B = -4$ by polynomial equality. Then we have $B = 1$ and $C = -3$, so we have
$$x^4-4x^2+x+2 = (x-1)(x^3+x^2-3x-2) = (x-1)(x+2)(x^2-x-1)$$
From here, you can find all $x$ and corresponding $y$'s by using the equations given (Don't forget to consider $x^2-x-1 = 0$ because it has two roots as well).
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