$a{x}^2+b{y}^2+c{z}^2+2ayz+2bzx+2cxy$  can be resolved into linear factors if $a,b,c$  are such that

$=z^2\left[a{\left(\frac{x}{z}\right)}^2+b{\left(\frac{y}{z}\right)}^2+c+2a\left(\frac{y}{z}\right)+2b\left(\frac{x}{z}\right)+2c\left(\frac{x}{z}\right)\left(\frac{y}{z}\right)\right]$

$=z^2\left[a{X}^2+b{Y}^2+c+2aY+2bX+2cXY\right]$

Where $\frac{x}{z}=X\text{and}\frac{y}{z}=Y$

Expression (1) will have two rational linear factors $x,y$  and $z$ , if expression $a{X}^2+b{Y}^2+2cXY+2bX+2aY+c$  will have two linear factors, if $abc+2abc-a.a^2-b.b^2-c.c^2=0$   or  $a^3+b^3+c^3=3abc$