<s>[INST] How prove infinitely many postive integers triples $(x,y,z)$ such $(x+y+z)^2+2(x+y+z)=5(xy+yz+zx)$ 
show that there exsit infinitely many  postive integers triples $(x,y,z)$
such
$$(x+y+z)^2+2(x+y+z)=5(xy+yz+zx)$$

May try it is clear $(x,y,z)=(1,1,1)$ is one  solution，and
$$(x+y+z+1)^2=5(xy+yz+xz)+1$$
 [/INST] For the equation.
$$(x+y+z)^2+2(x+y+z)=5(xy+xz+yz)$$
It is possible to reduce the parameterization of the solutions to some equivalent to the Pell equation.
It has the form.
$$x=3a^2-(b+c)a+b^2-3bc+c^2$$
$$y=a^2-(b+3c)a+3b^2-bc+c^2$$
$$z=a^2-(3b+c)a+b^2-bc+3c^3$$
These parameters can be recorded through the solution of the equation Pell.
$$p^2-5(2k^2+t^2)s^2=-1$$
$$a=ks$$
$$b=p-(3k-t)s$$
$$c=p-(3k+t)s$$ 
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