Let $k\geq-3$ be a fixed real number. Find the number of the solutions over reals of the following system: \begin{cases}a+b+c+d=3+k, \\ a^2+b^2+c^2+d^2=3+k^2, \\ abcd=k. \end{cases}
Similar problems (in order of apparition): http://gaceta.rsme.es/abrir.php?id=1432 , https://cms.math.ca/crux/v44/n2/Problems_44_2.pdf , etc... There are plenty more of them, none two of which similar.
What I thought: As for the system of equations, we need to start it from either $a^2+b^2$ as in $(a+b)^2 -2ab$ and same for $c$ and $d $ and then plot $k$ into as $abcd$ this will minus out one unknown but then we need to simplify with the first equation.