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...

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.
