<s>[INST] How many rational solutions does $(x^2+4x+8)^2+3x(x^2+4x+8)+2x^2$ have? 
How many rational solutions does $(x^2+4x+8)^2+3x(x^2+4x+8)+2x^2$ have?

I don't know how to start...
 [/INST] We can still use the classic "complete square" trick to do this without having to use the "double" variables as shown by the first answer. To this end, we have: $\left((x^2+4x+8) + \dfrac{3x}{2}\right)^2 - \dfrac{9x^2}{4}+2x^2=0\implies \left((x^2+4x+8)+\dfrac{3x}{2}\right)^2 = \left(\dfrac{x}{2}\right)^2$. Now using the familiar formula $a^2 = b^2 \implies a = \pm b$ to settle the answer.
</s>