Solving a system of equations, three variables x,y,z here's a problem from this PRMO which I found online:
Three real numbers $x,y,z$ are such that $x^{2}+6y=-17$, $y^{2}+4z=1$, $z^{2}+2x=2$. What is the value of $x^{2}+y^{2}+z^{2}$?
Ok I am quite confused about what to do because simple manipulations like adding the three equations don't seem to work (or at least I don't see anything yet). It would be really helpful if someone could tell me the approach or point out what I'm missing. Would be great if you could just give a small hint first instead of the whole answer immediately. Thanks!

Hint: From the first equation we get
$$y=-\frac{1}{6}\left(17+x^2)\right)$$ plugging this in the second equation and solving this for $z$
$$z=\frac{1}{4}\left(1-\frac{1}{36}\left(17+x^2\right)^2\right)$$ so we get
$$\frac{1}{16}\left(1-\frac{1}{36}(17+x^2)^2\right)^2+2x=2$$
Factorizing we obtain
$$\frac{(x+1)^2 \left(x^2+4 x+31\right) \left(x^4-6
   x^3+64 x^2-210 x+727\right)}{20736}=0$$
