Non-vanishing of this ternary quadratic expression [closed]

Question

I'm dealing with the expression $x^2+y^2+6z^2+8xy+4x+4y−6xz−6yz$. I want to show that this expression is always non-zero whenever $x,y$ and $z$ are positive integers. How does one do this? (Note that it's not always positive, e.g. $x=1,y=20,z=9$.)

Answer 1

your polynomial is zero for $x=1, y=19, z=9$

jagy@gost:~/Desktop/Cplusplus$ ./mse |    grep " 0"
Mon 20 Mar 2023 05:24:30 PM PDT
    x    y    z     poly 
    1   19    9    0
   19    1    9    0
    1   19   11    0
   19    1   11    0
    1   23    9    0
   23    1    9    0
    1   23   15    0
   23    1   15    0
    2   30   14    0
   30    2   14    0
    1   35   11    0
   35    1   11    0
    2   30   18    0
    2   34   14    0
   30    2   18    0
   34    2   14    0
    2   34   22    0
   34    2   22    0
    1   35   25    0
   35    1   25    0
    3   41   19    0
   41    3   19    0
    3   45   19    0
   45    3   19    0
    3   41   25    0
   41    3   25    0
    1   55   15    0
   55    1   15    0
    3   45   29    0
   45    3   29    0
    2   58   18    0
   58    2   18    0
    4   52   24    0
   52    4   24    0
    4   56   24    0
   56    4   24    0
    4   52   32    0
   52    4   32    0
    4   56   36    0
   56    4   36    0
    1   55   41    0
    5   63   29    0
   55    1   41    0
   63    5   29    0
    5   67   29    0
   67    5   29    0
    2   58   42    0
    2   78   22    0
   58    2   42    0
   78    2   22    0
    5   63   39    0
   63    5   39    0
    3   81   25    0
   81    3   25    0
    6   74   34    0
   74    6   34    0
    5   67   43    0
   67    5   43    0
    6   78   34    0
   78    6   34    0
Mon 20 Mar 2023 05:24:30 PM PDT

Answer 2

For $x=1$, $y = Y + 3 Z + 9$ and $z = Z+5$ we get the Pell-type equation $$ Y^2 - 3 Z^2 + 44 = 0 $$ which has infinitely many positive integer solutions.