<s>[INST] When does $nx^4+4x+3=0$ have real roots? 
Find all positive integers $n$ such that the equation $nx^4+4x+3=0$ has real roots. 

I think the answer must also include the cases with $2$ real roots. But my main question is, how do I start? Thanks.
 [/INST] We have
\begin{align}
nx^4+4x+3 & = (n-1)x^4 + x^4+4x+3 = (n-1)x^4 + (x+1)^2((x-1)^2+2)\\
& = (n-1)x^4 + (x^2-1)^2 + 2(x+1)^2
\end{align}
Hence, for $n \geq 2$, the function is a sum of squares and is always positive, since all three of them cannot be simultaneously zero and therefore has no real roots.
For $n=1$, we see that $x=-1$ is the only possible real root.
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