<s>[INST] $\begin{cases} x^4+x^2y^2+y^4=21 \\ x^2+xy+y^2=3 \end{cases}$ 
I should solve the following system: $$\begin{cases} x^4+x^2y^2+y^4=21 \\
 x^2+xy+y^2=3 \end{cases}$$
  by reducing the system to a system of second degree. 

What can I look for in such situations? What is the way to solve this kind of systems? The only thing I see here is that we can factor:
$$\begin{cases} x^2(x^2+y^2)+y^4=21 \\
 x(x+y)+y^2=3 \end{cases}$$
 [/INST] Hint:
$$x^4+y^4+x^2y^2=(x^2+y^2)^2-(xy)^2=?$$
So, we know $x^2+y^2,xy=9>0$
So, $x,y$ will have the same sign
Hope you can take it  from here using $y=\dfrac9x$ 
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